Research ArticleDiscussion of the Improved Methods for Analyzing a CantileverBeam Carrying a Tip-Mass under Base Excitation

Wang Hongjin1 Meng Qingfeng1 and Feng Wuwei2

1 Theory of Lubrication and Bearing Institute Xirsquoan Jiaotong University No 28 Xianning West Road Xirsquoan Shaanxi 710049 China2 School of Ship and Ocean Engineering Zhejiang Ocean University No 18 Haiyuan Road Zhoushan Zhejiang 316000 China

Correspondence should be addressed to Wang Hongjin whjin1984126com

Received 27 December 2012 Accepted 24 May 2013 Published 27 April 2014

Academic Editor Reza Jazar

Copyright copy 2014 Wang Hongjin et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Two improved analytical methods of calculations for natural frequencies and mode shapes of a uniform cantilever beam carryinga tip-mass under base excitation are presented based on forced vibration theory and the method of separation of variablesrespectively The cantilever model is simplified in detail by replacing the tip-mass with an equivalent inertial force and inertialmoment acting at the free end of the cantilever based on DrsquoAlembertrsquos principle The concentrated equivalent inertial forceand inertial moment are further represented as distributed loads using Dirac Delta Function In this case some typical naturalfrequencies and mode shapes of the cantilever model are calculated by the improved and unimproved analytical methods Thecomparing results show that after improvement these two methods are in extremely good agreement with each other even theoffset distance between the gravity center of the tip-mass and the attachment point is large As further verification the transientand steady displacement responses of the cantilever system under a sine base excitation are presented in which two improvedmethods are separately utilized Finally an experimental cantilever system is fabricated and the theoretical displacement responsesare validated by the experimental measurements successfully

1 Introduction

Laura et al [1] successfully determined the first ten naturalfrequencies of a clamped-free beam with a finite mass at thefree end using the standardmethod of separation of variablesRama Bhat and Wagner [2] considered the frequencies of auniform cantilever with an end mass using a power seriesexpansion [3] extended the research and presented thenatural frequencies and mode shapes of a cantilever beamwith a tip-mass and a base excitation by forced vibrationtheory The consistencies of the analysis results derived bytwo different analytical methods separately from [1 3] werediscussed in [4 5] Finally Jacquot [6] concluded and provedthat the results of the mode shapes of the cantilever beamcarrying a tipmass were both correct [1 3] RecentlyMousaviLajimi and Heppler [7] provided a further discussion on [2]and made the method more complete Soon Bhat [8] gavea response to the author of [7] On the other hand basedon the model from [3] Esmailzadeh and Nakhaie-Jazar [9]derived the equation of vibrationmotion of a cantilever beam

with a lumped mass while being excited harmonically atthe base Abramovich and Hamburger [10] further extendedthe research through the theory of Timoshenko beam Inrecent this model has been used for the application ofvibration control [11] and energy harvesting [12] Howeverthe conclusions and evidences above can only prove theconsistency of themethods of forced vibration and separationof variables for analyzing the cantilever beam when thegravity center of the tip-mass coincided with the point of theend attachment

In this paper we reinforced the method of separation ofvariables from [1] and developed the forced vibrationmethodfrom [3] Then the natural frequencies and mode shapesof a cantilever with an offset end-mass under transverseexcitation are derived using both of the improved analyticalmethods In addition the tip-mass is equated by an inertialforce and inertial moment acting at the free end of thecantilever beam based on DrsquoAlembertrsquos principle and theequivalent process is detailed The external loads includ-ing the concentrated inertia force the concentrated inertia

Hindawi Publishing CorporationShock and VibrationVolume 2014 Article ID 981053 15 pageshttpdxdoiorg1011552014981053

2 Shock and Vibration

b(t)

y

ua(x t)

xCg

o998400

rc

EI m L

M J

Figure 1 The model of cantilever beam with a tip-mass under baseexcitation

moment and the base excitation are represented as dis-tributed loads using Dirac Delta Function In this case sometypical natural frequencies and mode shapes are calculatedby the developed and undeveloped methods The comparingresults show that the inconsistency between the mode shapesderived by the undeveloped method become larger withthe increasing offset distance However after improvementthe mode shapes derived by two developed methods are inextremely good agreement with each other even the offsetdistance between the gravity center of the tip-mass andthe attachment point is large As further verification thetransient and steady displacement responses of the cantileversystem under a sine base excitation are presented by modalanalysis method in which the mode shapes derived by twoimproved analytical methods above are separately utilizedFinally an experimental cantilever system is fabricated andthe theoretical displacement responses are validated by theexperimental measurements successfully

2 The Mathematical Model

The uniform cantilever beam carrying a tip-mass at the freeend under base excitation is shown schematically in Figure 1It should be noticed that the gravity center of the tip-massis collinear with the gravity center of beam but does notcoincide with the point of the end attachment 1199001015840 The massand the moment of inertia of the tip-mass are both taken intoaccount but the rotary inertia of the beam is neglected

21 The Simplified Model of the Cantilever Beam with a Tip-Mass According to DrsquoAlembertrsquos principle the tip-mass atthe free end of the cantilever can be equated by an inertialforce and an inertial moment while vibrating In additionthe tip-mass with larger elastic modulus is always chosencomparing to the elastic modulus of beam So during vibra-tion the deformation of tip-mass can be ignored comparingto the beam That implies that the tip-mass can be regardedas a rigid body while vibrating In this case considering thetranslational motion and rotational motion about a fixed axisof rigid body the tip-mass can be equated by an inertialforce and an inertial moment The details about these twoequivalent processing will be discussed as follows

In the following analysis we will limit our study about thetip-mass to planar kinetics because the rigid tip-mass can beconsidered to be symmetrical with respect to a fixed referenceplane

First considering the translational effect since themotion of the rigid body can be viewed within the referenceplane all the forces and couple moments acting on the bodycan then be projected into the plane All particles insidethe tip-mass may move with the same acceleration (seeFigure 2(a)) The results of the simplification of this inertialforce system to the point 1199001015840 under translational motion givean inertial force and an inertial moment as follows

F(1)119892= sum F(1)

119892119894= minussum119898

119894a119894= minussum119898

119894a119888= minus119872a

119888

M(1)119892= sum r

119894times F(1)119892119894= minussum r

119894119898119894times a119888= minus119872r

119888times a119888

(1)

where the vector a119888indicates the acceleration of the gravity

center of the tip-mass the parameter 119872 indicates the valueof the tip-mass and r

119888presents the offset distance

Next in the condition of rotational motion about an axisperpendicular to the reference plane and passing throughpoint 1199001015840 all particles inside the rigid tip-massmay rotate withthe same angular acceleration (see Figure 2(b)) Reducing theinertial rotating system to point 1199001015840 gives the other inertialforce and inertial moment as

F(2)119892= sum F(2)

119892119894= minussum119898

119894a120591119894= minussum119898

119894(120585 times r

119894) = minus119872120585 times r

119888

M(2)119892= sum r

119894times F(2)119894= minussum119898

119894(r119894times 120585 times r

119894) = minus120585sum119898

1198941199032

119894

= minus 120585 (119869 +1198721199032

119888)

(2)

where 120585 and 119869 indicate the angular acceleration and themoment of inertia of the rigid tip-mass respectively

In this case the total inertial force and inertial momentof the tip-mass can be written as the sum of the inertialequivalent forces and moments from translational and rota-tional motions Assuming that the acceleration and angularacceleration of all particles inside of tip-mass are equal tothe accelerations of the particle at point 1199001015840 then the inertialequivalent force and moment of the rigid tip-mass would bewritten in scalar quantity as

119865119892= 119872

1205972119906119886(119871 119905)

1205971199052

+ 119903119888119872

1205972

1205971199052[

120597119906119886(119871 119905)

120597119909

]

119872119892= 119903119888119872

1205972119906119886(119871 119905)

1205971199052

+ (119869 + 1199032

119888119872)

1205972

1205971199052[

120597119906119886(119871 119905)

120597119909

]

(3)

where 119906119886(119871 119905) indicates the absolute transverse displacement

at the free end of the cantilever beam The directions of thewhole inertial equivalent force and moment can be foundaccording to Newtonrsquos third law of motion as shown inFigure 3

22 The Reinforced Method of Separation of Variables Inthis section the natural frequencies and mode shapes of thecantilever model (as shown in Figure 3) will be derived bythe reinforced method of separation of variables First usingDiracDelta Function [13] the concentrated equivalent inertia

Shock and Vibration 3

ri

rcCg

mi

aiac

F(1)gi

F(1)g

o998400

M(1)g

(a)

ri

rcCg

mi

acF(2)gi

F(2)g

o998400

a120591i

M(2)g

120585

(b)

Figure 2 Simplification of the tip-mass to point 1199001015840 under (a) translational motion and (b) rotational motion

y

x

b(t) ua(x t)

FgMg

EI m L

Figure 3 The simplified model of cantilever with tip-mass beingequated by inertial force and moment

force 119865119892and inertia moment 119872

119892can be represented in the

form of distributed loads as

119891119886(119909 119905)

= 119865119892120575 (119909 minus 119871)

= [119872

1205972119906119886(119909 119905)

1205971199052

+ 119903119888119872

1205972

1205971199052(

120597119906119886(119909 119905)

120597119909

)] 120575 (119909 minus 119871)

119872119886(119909 119905)

= 119872119892120575 (119909 minus 119871)

= [119903119888119872

1205972119906119886(119909 119905)

1205971199052

+ (119869 + 1199032

119888119872)

1205972

1205971199052(

120597119906119886(119909 119905)

120597119909

)]

times

119889120575 (119909 minus 119871)

119889119909

(4)

Therefore employing variational method the dynamicmotion equation of the simplified model of the cantileverbeam with a tip-mass under base transverse excitation isobtained as

119864119868

1205974119906119886(119909 119905)

120597119909

+ 119898

1205972119906119886(119909 119905)

120597119905

= 119891119886(119909 119905) minus 119872

119886(119909 119905) (5)

where 119864119868 and119898 denote the flexural rigidity and the mass perunit length of the beam respectively

The absolute displacement can be considered as thesum of the relative displacement and the base displacementexcitation

119906119886(119909 119905) = 119906rel (119909 119905) + 119887 (119905) (6)

Substituting (6) into (5) gives the new form of the dynamicmotion equation as

119864119868

1205974119906rel (119909 119905)

120597119909

+ 119898

1205972119906rel (119909 119905)

120597119905

= [119872

1205972119906rel (119909 119905)

1205971199052

+ 119903119888119872

1205972

1205971199052(

120597119906rel (119909 119905)

120597119909

)] 120575 (119909 minus 119871)

+ [119903119888119872

1205972119906rel (119909 119905)

1205971199052

+ (119869 + 1199032

119888119872)

1205972

1205971199052(

120597119906rel (119909 119905)

120597119909

)]

times

119889120575 (119909 minus 119871)

119889119909

+ [119872120575 (119909 minus 119871) + 119903119888119872

119889120575 (119909 minus 119871)

119889119909

minus 119898]

1198892119887 (119905)

1198891199052

(7)

In this case the simplified model of the cantilever beamunder base excitation can be considered as a new analyticalmodel with the dynamic differential equation of (8) and theboundary conditions of (9) through (12)

119864119868

1205974119906119886(119909 119905)

120597119909

+ 119898

1205972119906119886(119909 119905)

120597119905

= [119872120575 (119909 minus 119871) + 119903119888119872

119889120575 (119909 minus 119871)

119889119909

minus 119898]

1198892119887 (119905)

1198891199052

(8)

at 119909 = 0

119906119886(119909 119905) = 0 (9)

119889

119889119909

119906119886(119909 119905) = 0 (10)

4 Shock and Vibration

at 119909 = 119871

119864119868

1198893

1198891199093119906119886(119909 119905) = [119872

1205972119906119886(119909 119905)

1205971199052

+ 119903119888119872

1205972

1205971199052(

120597119906119886(119909 119905)

120597119909

)]

(11)

119864119868

1198892

1198891199092119906119886(119909 119905)

= minus [119903119888119872

1205972119906119886(119909 119905)

1205971199052

+ (119869 + 1199032

119888119872)

1205972

1205971199052(

120597119906119886(119909 119905)

120597119909

)]

(12)

The solutions of (8) subjected to four boundary condi-tions and two initial conditions can be obtained convenientlyby the traditional method of separation of variables Themethod regards the response as a superposition of the systemeigen-functionsmultiplied by corresponding time dependentgeneralized coordinates Of course this necessitates firstobtaining the solution of the system eigen-value problemTurning attentions to the corresponding eigen-value problemand considering the free vibration characterized by setting119887(119905) = 0 the solution of (8) becomes separable in space andtime Letting

119906119886(119909 119905) = 119880 (119909) 119879 (119905) (13)

and substituting (13) into the free vibration form of (8) twodependent equations are obtained using the separation ofvariables method as

119889

1198891199052119879 (119905) + 120596

2119879 (119905) = 0 (14)

1198894

1198891199094119880 (119909) minus 120573

4119880 (119909) = 0 (15)

where 1205962 indicates the angular frequency and 1205734 =

1198981205962(119864119868) In this case the four boundary conditions will be

changed as

at 119909 = 0

119880 (119909) = 0 (16)

119889

119889119909

119880 (119909) = 0 (17)

at 119909 = 119871

119864119868

1198893

1198891199093119880 (119909) = minus [120596

2119872119880(119909) + 119903

1198881198721205962(

119889

119889119909

119880 (119909))] (18)

119864119868

1198892

1198891199092119880 (119909) = 119903

1198881198721205962119880 (119909) + (119869 + 119903

2

119888119872)1205962(

119889

119889119909

119880 (119909))

(19)

The general solution of (15) can be easily derived to be

119880 (119909) = 1198621cos (120573119909) + 119862

2sin (120573119909) + 119862

3cosh (120573119909)

+ 1198624sinh (120573119909)

(20)

where 119862119899(119899 = 1 2 3 4) are constants of integration In

order to evaluate three of these constants in terms of thefourth and derive the characteristic equation the boundaryconditions (16) to (19) must be used Indeed substituting (20)into (16) through (19) gives the characteristic equation andthe mode shapes as

1 + 119888119888ℎ minus 120601119906 (119904119888ℎ minus 119888119904ℎ)

minus 1198771[119904119888ℎ + 119888119904ℎ minus 120601119906 (1 minus 119888119888ℎ)]

minus 21198772119904119904ℎ minus 119877

2

2(1 minus 119888119888ℎ) = 0

(21)

119880 (119909) = 1198622sin (120573119909) minus 119904ℎ (120573119909) minus 120582 [cos (120573119909) minus 119888ℎ (120573119909)]

(22)

where 120582 = 1198601211986011and the constant 119862

2is an arbitrary value

and in which

11986011= minus(1 + 119877

2)119888 + (119877

2minus 1)119888ℎ + 119877

1(119904 + 119904ℎ) 119860

12=

1198771(119888ℎ minus 119888) minus (1 + 119877

2)119904 + (119877

2minus 1)119904ℎ

119906 = 120573119871 119904 = sin(119906) 119888 = cos(119906) 119904ℎ = sinh(119906) 119888ℎ =cosh(119906)

120601 = 119872(119898119871) 1198771= 120601119906

3[(119870119871)

2+ (119903119888119871)2] 1198772=

1206011199062(119903119888119871) 119870 = radic119869119872

23TheDevelopedMethod of Forced Vibration If we treat thebase displacement excitation as a kind of boundary conditionthen the differential equation of motion and four boundaryconditions of the cantilever system can be written as

119864119868

1205974119906119886(119909 119905)

120597119909

+ 119898

1205972119906119886(119909 119905)

120597119905

= 0 (23)

at 119909 = 0

119906119886(0 119905) = 119887 (119905) (24)

119889

119889119909

119906119886(0 119905) = 0 (25)

at 119909 = 119871

119864119868

1198893

1198891199093119906119886(119909 119905)

= [119872

1205972119906119886(119909 119905)

1205971199052

+ 119903119888119872

1205972

1205971199052(

120597119906119886(119909 119905)

120597119909

)]

(26)

119864119868

1198892

1198891199092119906119886(119909 119905)

= minus [119903119888119872

1205972119906119886(119909 119905)

1205971199052

+ (119869 + 1199032

119888119872)

1205972

1205971199052(

120597119906119886(119909 119905)

120597119909

)]

(27)

Shock and Vibration 5

Under the simple harmonic excitation 119890119895120596119905 the steady statesolution of (23) can be regarded with the form as

119906119886(119909 119905) = 119880

119886(119909 120596) 119890

119895120596119905minus120593

= [1198611cos (120573119909) + 119861

2sin (120573119909)] 119890119895120596119905minus120593

+1198613cosh (120573119909) + 119861

4sinh (120573119909)]

(28)

where 119861119899(119899 = 1 2 3 4) are arbitrary constants the

parameter 120573 is defined by 1205734 = 1198981205962(119864119868) 119895 is the imaginaryunit and 120593 is the lagging phase

We recalculated the natural frequencies and the modeshapes directly by substituting (28) into four boundaryconditions (24) through (27) and obtained the characteristicequation and displacement response which are different from[3]

119863 = 1 + 119888119888ℎ minus 120601119906 (119904119888ℎ minus 119888119904ℎ)

minus 1198771[119904119888ℎ + 119888119904ℎ minus 120601119906 (1 minus 119888119888ℎ)]

minus 21198772119904119904ℎ minus 119877

2

2(1 minus 119888119888ℎ)

119880 (119909) = (

119861

2119863

) 1198651015840

1119906cos (120573119909) + 1198651015840

2119906cosh (120573119909)

+1198651015840

3119906[sin (120573119909) minus sinh (120573119909)]

(29)

where 119861 is defined by 119887(119905) = 119861119890119895120596119905 and in which

1198651015840

1119906= 1+119888119888ℎminus119904119904ℎminus2120593119906119904119888ℎminus2119877

1119888119904ℎ+(119877

1120593119906minus119877

2

2)(1minus

119888119888ℎ + 119904119904ℎ) minus 21198772(119888119888ℎ + 119904119904ℎ) =119865

1119906minus 21198772(119888119888ℎ + 119904119904ℎ)

1198651015840

2119906= 1+119888119888ℎ+119904119904ℎ+2120593119906119888119904ℎminus2119877

1119904119888ℎ+(119877

1120593119906minus119877

2

2)(1minus

119888119888ℎ minus 119904119904ℎ) + 21198772(119888119888ℎ minus 119904119904ℎ) =119865

2119906+ 21198772(119888119888ℎ minus 119904119904ℎ)

1198651015840

3119906= (119904119888ℎ + 119888119904ℎ)(1 minus 119877

1120593119906 +119877

2

2) + 2[120593119906119888119888ℎ minus 119877

1119904119904ℎ minus

1198772(119904119888ℎ minus 119888119904ℎ)] = 119865

3119906

It is obvious that the characteristic equation derived by thereinforced method of separation of variables is equal tothe equation obtained by the developed method of forcedvibration

Themode shapes cannot be obtained directly because theboundary conditions contain the parameter1205962 However thisproblem can be successfully solved by Dirac Delta Function[13] Thus the mode shapes may be obtained by the Sturm-Liouville theory [14] and expressed as

119880119894(119909) = (

1

2

) 1198651015840

1119906119894119865 cos (120573

119894119909) + 119865

1015840

2119906119894cosh (120573

119894119909)

+ 1198651015840

3119906119894[sin (120573

119894119909) minus sinh (120573

119894119909)]

(30)

3 Computed Results and Comparisons

In this section three types of the natural mode shapes ofthe cantilever beam with a tip-mass were computed and

compared The first one is called the undeveloped modeshapes which is directly obtained from [3] by the expression

119880(1)

119894(119909) = (

1

2

) 1198651119894119888

cos(119906119894119909

119871

) + 1198652119894119888

cosh (119906119894119909

119871

)

+ 1198653119894119888[sin(

119906119894119909

119871

) minus sinh(119906119894119909

119871

)]

(31)

where 1198651119894119888

= 1198651119894119888ℎ(119906119894) 1198652119894119888

= 1198652119894119888ℎ(119906119894) and 119865

3119894119888=

1198653119894119888ℎ(119906119894) in which 119865

1119894 1198652119894 and 119865

3119894are the functions 119865

1119906 1198652119906

and 1198653119906

with the variable 119906 replaced by 119906119894 respectively

The second one is based on the expression of thedeveloped mode shapes derived in this paper Let (30) bemultiplied by 21198651015840

3119894to give the second mode shapes as

119880(2)

119894(119909) = sin(

119906119894119909

119871

) minus sinh(119906119894119909

119871

) + (

1198651015840

1119894

1198651015840

3119894

) cos(119906119894119909

119871

)

+ (

1198651015840

2119894

1198651015840

3119894

) cosh (119906119894119909

119871

)

(32)

At last the arbitrary constant 1198622in (22) was chosen as

constant 1 to give the third type of the natural mode shapes as

119880(3)

119894(119909) = sin(

119906119894119909

119871

) minus 119904ℎ (

119906119894119909

119871

)

minus 120582 [cos(119906119894119909

119871

) minus 119888ℎ (

119906119894119909

119871

)]

(33)

Before the mode shapes are determined the roots ofthe characteristic equation must be calculated firstly Thecharacteristic equation (21) or 119863 = 0 can be solved by twosteps First approximate roots were pointed out by applyinggraphing method Then the approximate roots were iteratedby the bisection method to give the exact characteristic rootswith given error All these processes were finished usingMATLAB software

Because the condition that the gravity center of the tip-mass coincided with the point of the end attachment hasbeen verified in [6] we focus on the condition when theoffset distance is not zero Table 1 shows the first five roots ofthe characteristic equations from five samples with differentvalues of 120601 119896119871 and 119889119871 It can be clearly noticed thatthe larger value of 119889119871 resulted in the less value of the firstcharacteristic root but the larger values of the other fourcharacteristic roots when 120601 and 119896119871 were set as constants

Once the roots of the characteristic equation have beenobtained three types of the first five mode shapes of thecantilever structure from the five given samples can becalculated Figures 4 5 6 7 and 8 show the first five modeshapes of the cantilever with various combinations of 120601 119896119871and 119889119871

6 Shock and Vibration

Table 1 Typical roots of characteristic equation with variables 120601 119896119871 and 119889119871

Sample 120601 119870119871 119889119871

Roots of characteristic equation1199061

1199062

1199063

1199064

1199065

1 4 4 0 035211 129234 477818 788401 11017802 4 4 05 034814 130637 477952 788472 11018273 4 4 1 034200 132825 478225 788634 11019404 4 4 2 032544 138983 479177 789233 11023685 4 4 4 028806 154543 482543 791455 1103990

003002001

0

0 05 1 0 05 1 0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

000000

050000

100000

i = 1

000000

050000

100000

i = 1

000000 000000

000000

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

000000

050000

100000

i = 2

000000

050000

100000

i = 2

minus186571 times 10minus9

000792

003152

minus198560 times 10minus8

minus9418658

minus18545378

0

minus100

minus2000 05 1

0 05 1

0 05 1

0 05 1

1050

times104 i = 3

000000

050000

100000

i = 3

000000

050000

100000

i = 3

786191 times 10minus6

111026 times 104

minus686262 times 103

50

minus5

times105 i = 4000000

050000

100000

i = 4

000000

050000

100000

i = 4

i = 5000000

050000

100000

i = 5

000000 000000

050000

100000

i = 5times106

20

minus2

000017

minus271300 times 106

minus855944 times 104

1

05

0

1

05

0

03

02

01

0

03

02

01

0

151

050

151

050

2

0

minus2

2

0

minus2

210

minus1

210

minus1

minus493414 times 10minus8

032943

131050

367824 times 10minus11

017448

034354

113682 times 10minus10

160541

minus009923

457130 times 10minus11

minus002114

006144

866744 times 10minus11

minus140614

minus004436

032943

131050

017448

034354

160541

minus009923

000000

minus002114

006144

minus140614

minus004436

Notes at xL Notes at xL Notes at xLValues ValuesValues

Ui(x)

xL

Ui(x)

xL

Ui(x)

xL

minus233509 times 10minus5

107963 times 104

minus313859 times 104

First-type mode shapes U(1)i

(x) Second-type mode shapes U(2)i

(x) Third-type mode shapes U(3)i

(x)

Figure 4 Three types of the first five mode shapes with the combinations of 120601 = 4 119896119871 = 4 and 119889119871 = 0

It is obvious that when 119889119871 = 0 the first type of naturemode shapes 119880(1)

119894(119909) derived by the undeveloped method

turned far away from the third-type mode shapes 119880(3)119894(119909)

derived by the reinforced method of separation of variablesIn addition the larger value of 119889119871 resulted in the moreinconsistent mode shapes However the second type of modeshapes 119880(2)

119894(119909) obtained by the developed method of forced

vibration seemed in good agreement with the third onesFurthermore the amplitude of the first mode shape

decreased but the other four mode shapes increased with theincreasing value of the 119889119871 when 119896119871 and 120601 were fixed to be

constants The effect from the varying value of 119889119871 for thelower mode shapes was always stronger than for the higher

4 Experimental Verification

In order to experimentally explain and compare two typesof natural mode shapes derived by two improved methodsabove the transient and steady responses of the cantileversystem under an input base excitation should be calculatedfirst This whole calculation can be finished by method ofmodal analysis But above all the orthogonality property of

Shock and Vibration 7

0 05 1

0 05 1

Ui(x)

xL

Ui(x)

xL

Ui(x)

xL

014013012011

0

minus100

minus200

000000

050000

100000

i = 1

000000

050000

100000

i = 2

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

011044

011840

014143

533123

minus9287788

minus18666064

0 05 1

1050

times104

000000

050000

100000

i = 3minus8449356

107519 times 105

minus970528 times 103

0 05 1

50

minus5

times105

000000

050000

100000

i = 422493344

139452 times 104

139108 times 105

0 05 1

40

minus4minus8

times106

000000

050000

100000

i = 5minus42806116

minus266045 times 106

minus765307 times 106

0 05 1

000000

050000

100000

i = 106

04

02

0

392120 times 10minus9

017490

069258

0 05 1

06

04

02

0

0 05 1

000000

050000

100000

i = 2547973 times 10minus11

017881

034942

03

02

01

00 05 1

03

02

01

0

0 05 1

000000

050000

100000

i = 3minus505260 times 10minus11

160517

minus010133

151

050

0 05 1

151

050

0 05 1

000000

050000

100000

i = 4minus776048 times 10minus11

minus002162

006261

2

0

minus20 05 1

2

0

minus2

0 05 1

000000

050000

100000

i = 5663249 times 10minus11

minus140614

minus004516

210

minus1

000000

000000

000000

000000

017490

069258

017881

034942

160517

minus010133

minus002162

006261

000000

050000

100000

i = 5

0 05 1

210

minus1

000000

minus140614

minus004516

Notes at xL Notes at xL Notes at xLValues ValuesValuesFirst-type mode shapes U(1)i

(x) Second-type mode shapes U(2)i

(x) Third-type mode shapes U(3)i

(x)

Figure 5 Three types of the first five mode shapes with the combinations of 120601 = 4 119896119871 = 4 and 119889119871 = 05

034

033

032

031

0 05 1 0 05 1 0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1 0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0

minus100

minus200

10

5

0

times104

50

minus5

times105

times106

0minus5minus10minus15

Ui(x)

xL

Ui(x)

xL

Ui(x)

xL

04

02

0

04

02

0

04

02

0

04

02

0

151

050

151

050

2

0

minus2

2

0

minus2

210

minus1

210

minus1

000000

050000

100000

i = 1

000000 000000

050000

100000

i = 1

030899

031712

033934

000000

050000

100000

i = 2861965

minus9105604

minus18440224

000000

050000

100000

i = 3minus10637851

973291 times 104

minus123754 times 104

000000

050000

100000

i = 419388076

168672 times 104

311790 times 105

000000

050000

100000

i = 5minus19120922

minus244086 times 106

minus152554 times 107

000000

050000

100000

i = 1

331605 times 10minus9

012125

047836

000000

050000

100000

i = 2

minus550109 times 10minus11

018658

036219

000000

050000

100000

i = 3

minus175471 times 10minus11

160459

minus010631

000000

050000

100000

i = 4

109777 times 10minus10

minus002275

006562

000000

050000

100000

i = 5

836395 times 10minus11

minus140614

minus004731

012125

047836

000000 000000

050000

100000

i = 2

018658

036219

000000 000000

050000

100000

i = 4

minus002275

006562

000000 000000

050000

100000

i = 5

minus140614

minus004731

000000 000000

050000

100000

i = 3

160459

minus010631

Notes at xL Notes at xL Notes at xLValues ValuesValuesFirst-type mode shapes U(1)i

(x) Second-type mode shapes U(2)i

(x) Third-type mode shapes U(3)i

(x)

Figure 6 Three types of the first five mode shapes with the combinations of 120601 = 4 119896119871 = 4 and 119889119871 = 1

8 Shock and Vibration

088087086085

0 05 1 0 05 1 0 05 1

0 05 1 0 05 1

0 05 10 05 1

0 05 1 0 05 1

0 05 1 0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0

minus50

minus100

minus150

6420

minus2

times104

840

minus4

times105

0minus1minus2minus3

times107

0302010

03

02

01

0

04

02

0

04

02

0

151

050

151

050

210

minus1

210

minus1

210

minus1

210

minus1

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

084906

085736

087731

minus1308377

minus9308045

minus16072843

63490847

565642 times 104

minus172997 times 104

minus329762092

230021 times 104

667852 times 105

946242416

minus149912 times 106

minus306922 times 107

minus995048 times 10minus10

007811

030669

511030 times 10minus11

021118

040600

minus114494 times 10minus10

160244

minus012466

688252 times 10minus11

minus002686

007708

183045 times 10minus11

minus140613

minus005561

000000

000000

000000

000000

000000

007811

030669

021118

040600

160244

minus012466

minus002686

007708

minus140613

minus005561

Ui(x)

xL

Ui(x)

xL

Ui(x)

xL

Notes at xL Notes at xL Notes at xLValues ValuesValuesFirst-type mode shapes U(1)i

(x) Second-type mode shapes U(2)i

(x) Third-type mode shapes U(3)i

(x)

Figure 7 Three types of the first five mode shapes with the combinations of 120601 = 4 119896119871 = 4 and 119889119871 = 2

the natural mode shapes should be derived In fact usingBetti theorem [15] the orthogonality properties with respectto mass and stiffness can be obtained respectively as follows

int

119871

0

119898119880(119896)

119903(119909)119880(119896)

119894(119909) 119889119909 +119872119880

(119896)

119903(119871)119880(119896)

119894(119871)

+ 119903119888119872119880(119896)

119894(119871)

119889119880(119896)

119903(119871)

119889119909

+ 119903119888119872119880(119896)

119903(119871)

119889119880(119896)

119894(119871)

119889119909

+ (119869 + 1199032

119888119872)

119889119880(119896)

119894(119871)

119889119909

119889119880(119896)

119903(119871)

119889119909

= (120575(119896)

119903119894)

2

(119896 = 1 2 119903 119894 = 1 2 )

(34)

int

119871

0

119864119868

1198892119880(119896)

119894(119909)

1198891199092

1198892119880(119896)

119903(119909)

1198891199092

119889119909 = (120596119903120575(119896)

119903119894)

2

(119896 = 1 2 119903 119894 = 1 2 )

(35)

In this case the normalized mode shapes can be defined as

119880(119896)

119903119873(119909) =

119880(119896)

119903(119909)

120575(119896)

119903119903

(36)

where

(120575(119896)

119903119903)

2

= int

119871

0

119898[119880(119896)

119903(119909)]

2

119889119909 +119872[119880(119896)

119903(119871)]

2

+ 2119903119888119872119880(119896)

119903(119871) [

119889

119889119909

119880(119896)

119903(119871)]

+ (119869 +1198721199032

119888) [

119889

119889119909

119880(119896)

119903(119871)]

2

(37)

41 Theoretical Responses of the Cantilever System with a BaseSine Excitation According to the method of modal analysisthe solution of (8) has the form of

119906119886(119909 119905) =

infin

sum

119903=1

119880(119896)

119903119873(119909) 119902119903(119905) (38)

So inserting (38) into (8) gives

119864119868

infin

sum

119903=1

1198894119880(119896)

119903119873(119909)

1198891199094

119902119903(119905) + 119898

infin

sum

119903=1

119880(119896)

119903119873(119909)

1198892119902119903(119905)

1198891199052

= [119872120575 (119909 minus 119871) + 119903119888119872

119889120575 (119909 minus 119871)

119889119909

minus 119898]

1198892119887 (119905)

1198891199052

(39)

Considering the orthogonality properties of equations (34)and (35) let (39) be multiplied by119880(119896)

119894119873(119909) and integrated over

Shock and Vibration 9

192

191

19

0 05 1 0 05 1

0 05 1 0 05 1

0 05 1

0 05 10 05 1

0 05 10 05 1

0 05 1 0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0

minus200

minus400

minus600

0minus5minus10

times104

1

0

minus1

times106

times107

0minus2minus4minus6

02

01

0

02

01

0

06

04

02

0

06

04

02

0

15

1

05

0

15

1

05

0

210

minus1

210

minus1

210

minus1

210

minus1

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

000000

050000

100000

i = 1

000000

050000

100000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

189913

190621

192051

minus56420429

minus28138349

4723398

134513 times 104

minus110859 times 105

minus260952 times 104

minus615688 times 104

496921 times 104

145091 times 106

168417 times 105

241806 times 106

minus633899 times 107

137503 times 10minus9

004964

019458

minus238241 times 10minus11

028530

054402

159443

minus019150

minus283947 times 10minus11

minus004212

012050

771494 times 10minus11

minus140605

minus008755

000000

000000

000000 000000

000000

000000

004964

019458

028530

054402

159443

minus019150

minus004212

012050

minus140605

minus008755

Ui(x)

xL

Ui(x)

xL

Ui(x)

xL

Notes at xL Notes at xL Notes at xLValues ValuesValues

minus387890 times 10minus11

First-type mode shapes U(1)i

(x) Second-type mode shapes U(2)i

(x) Third-type mode shapes U(3)i

(x)

Figure 8 Three types of the first five mode shapes with the combinations of 120601 = 4 119896119871 = 4 and 119889119871 = 4

the domain 0 lt 119909 lt 119871 to give a set of independent ordinarydifferential equations as

1198892119902119894(119905)

1198891199052

+ 1205962

119894119902119894(119905) = 119891

(119896)

119894(119905) (40)

Considering the properties of Dirac Delta Function

int

119871

0

119880(119896)

119894119873(119909) 120575 (119909 minus 119871) 119889119909 = 119880

(119896)

119894119873(119871)

int

119871

0

119880(119896)

119894119873(119909)

119889120575 (119909 minus 119871)

119889119909

119889119909 = minus

119889119880(119896)

119894119873(119871)

119889119909

(41)

the generalized forces associated with the generalized coordi-nates 119902

119894(119905) can be obtained as

119891(119896)

119894(119905)

= [119872119880(119896)

119894119873(119871) minus 119903

119888119872

119889119880(119896)

119894119873(119871)

119889119909

minus 119898int

119871

0

119880(119896)

119894119873(119909) 119889119909]

1198892119887 (119905)

1198891199052

(42)

Furthermore considering the mechanical damping asRayleigh damping and ignoring the part of mass dampingthe damping coefficient 120577 of the bimorph beam can be deter-mined by the modal damping ratio which can be obtained

by experiment [16] The independent ordinary differentialequations with damping then can be written as

1198892119902119894(119905)

1198891199052

+ 2120596119894120577

119889119902119894(119905)

119889119905

+ 1205962

119894119902119894(119905)

= [119872119880(119896)

119894119873(119871) minus 119903

119888119872

119889119880(119896)

119894119873(119871)

119889119909

minus 119898int

119871

0

119880(119896)

119894119873(119909) 119889119909]

1198892119887 (119905)

1198891199052

(43)

Now (43) can be solved using Laplace transform Applyinga sine excitation 119887(119905) = 119861 sin(120596119905) to the base of cantileverstructure under zero initial conditions the transfer functionof the independent differential equations for the cantileverbeam with tip-mass under base displacement excitation canbe written as

119866119894(119904)

=

119876119894(119904)

119861119894(119904)

=

(minus1205962)[119872119880

(119896)

119894119873(119871)minus119903

119888119872(119889119880

(119896)

119894119873(119871) 119889119909)minus119898int

119871

0119880(119896)

119894119873(119909) 119889119909]

1199042+ 2120596119894120577119904 + 120596

2

119894

(44)

10 Shock and Vibration

where 119876119894(119904) and 119861

119894(119904) are the Laplace transforms of 119902

119894(119905) and

119887(119905) respectively The inverse transformation of the transferfunction will be further obtained as

119892119894(119905)

=

(minus1205962)[119872119880

(119896)

119894119873(119871)minus119903

119888119872(119889119880

(119896)

119894119873(119871) 119889119909)minus119898int

119871

0119880(119896)

119894119873(119909) 119889119909]

120596119894radic1 minus 120577

2

times 119890minus120596119894120577119905 sin(120596

119894radic1 minus 120577

2119905)

(45)

In this case the solutions of (43) can be solved by convolutiontheorem as

119902119894(119905) = 119887 (119905) lowast 119892

119894(119905) = int

119905

0

119887 (120591) 119892119894(119905 minus 120591) 119889120591 (46)

Finally substituting the result of (46) into (38) and after somelengthy algebraic manipulation the absolute displacementresponse of the cantilever beam with tip-mass under basetransverse excitation can be obtained as

119906119886(119909 120596 119905) =

infin

sum

119894=1

119880(119896)

119894119873(119909)

119861 [119872119880(119896)

119894119873(119871) minus 119903

119888119872(119889119880

(119896)

119894119873(119871) 119889119909) minus 119898int

119871

0119880(119896)

119894119873(119909) 119889119909]

radic1205964+ 2 (2120577

2minus 1) 120596

21205962

119894+ 1205964

119894

times [

1205963

120596119894radic1 minus 120577

2

119890minus120596119894120577119905 sin(120596

119894radic1 minus 120577

2119905 + 1206011) + 1205962 sin (120596119905 + 120601

2)]

(47)

where 1206011and 120601

2are determined by tan(120601

1) = 2120577radic1 minus 120577

2

((120596120596119894)2+ 21205772minus 1) and tan(120601

2) = minus2120577(120596120596

119894)(1 minus (120596120596

119894)2)

It can be seen that the solution consists of two parts abouttimeThe first term represents free vibration which will decaywith time exponentially The second term depending on theexcitation represents the forced vibration of the system

42 Experimental Responses and Comparisons with the Theo-retical Results In order to explain the absolute displacementresponse of the cantilever structure under base excitationexperimentally a typical cantilever beam in which theelongated beam was inserted and fixed into the tip-mass tomake sure the gravity center of tip-mass was collinear withthe beam was fabricated (as shown in Figure 9) The chosengeometric and physical properties of the cantilever beamincluding the tip-mass are given in Table 2

The complete experimental devices are shown inFigure 10 The electromagnetic vibrating table type SW-TFAcontrolled by a programmable controller type SW-F-2TF-2was used to create a specific harmonic input excitationThe cantilever beam with a tip-mass was clamped onthe vibrating top platform Eddy-Current Sensors typeDM6008 and ZA210800 were used to measure the inputdisplacement from base vibrating exciter and the absolutedynamic displacement from the beam respectively Allmeasurements from cantilever beam and vibrating exciterwere connected to Digital Oscilloscope type DS1204Band the Industrial Computer including a PC-based DataAcquisitionModule from ADLINK company type DAQ2213The Digital Oscilloscope and Industrial Computer displayedand processed the measurement results using the DAQPilotsoftware and the results from subsequent analysis using theMATLAB software

Table 2 Characteristic properties of the cantilever beam includingtip-mass

Physical properties Beam Tip-massYoungrsquos modulus 119864 (GPa) 69 mdashDensity 120588

119904or 120588119898(kgm3) 2630 7753

Length 119871 or 119871119898(mm) 1204 156

Thickness 119905119904or 119905119898(mm) 49 295

Width 119908 (mm) 102 102Total mass119872 (g)a mdash 3383Offset 119903

119888(mm)b mdash 801

Rotary inertia 119869 (kgsdotm2)b c mdash 333 times 10minus6aThe total mass of the tip-mass including the elongated part of beam wascalculated by119872 = 119871

119898119905119898119908120588119898+ 119871119904119905119904119908(120588119904minus 120588119898) where 119871

119904= 10mm

bThe offset distance and the rotary inertia were calculated according to thegeometry and material properties of the beam and tip-mass using equations119903119888= (120588

1198981199051198981198712

119898minus 1205881198981199051199041198712

119904+ 1205881199041199051199041198712

119904)(2[120588

119898119905119898119871119898+ 119871119904119905119904(120588119904minus 120588119898)])

119869 = 120588119898119871119898119905119898119908 [(112)(119871

2

119898+ 1199052

119898) + ((12)119871

119898minus 119903119888)2] + 119871

119904119905119904119908(120588119904minus

120588119898)[(112)(119871

2

119904+ 1199052

119904) + (119903119888minus (12)119871

119904)2]

cThe corresponding parameters 120601 119896119871 and 119903119888119871 were calculated as follows

120601 = 119872(120588119904119905119904119908) = 21377 119896119871 = radic119869119872119871 = 00824 and 119903

119888119871 = 00665

The analytical model defined by (47) in which two typesof mode shapes derived by two improved analytical methodsare utilized can be used to analyze the steady and transientdisplacement responses of the cantilever beam with a tip-mass and an input base excitation based on the chosenvarying frequencies time and positions along the lengthof the beam Typically the displacement responses frompositions along 119909-axis at 119909

1= 364mm 119909

2= 574mm

1199093

= 784mm 1199094

= 994mm and 1199095

= 1204mmoff the origin point (as shown in Figure 9) were measured

Shock and Vibration 11

y

b(t)

x1

x2

x3

x4

x5

Ls

L

t s

t m x

Lm

rc

Cg

o998400

Figure 9 Typical cantilever beam with a tip-mass

1 2

3

56

7

8

4

Tip-massCantilever beam

(7) Cantilever with a tip-mass

(1) Industrial computer including data acquisitionmodule type DAQ2213

(2) Digital oscilloscope type DS1204B(3) Programmable controller type SW-F-FTF-2(4) Power supply type DP1308A(5) Driver modules for Eddy-current sensors(6) Electromagnetic vibrating table type SW-TFA

(8) Eddy-current sensors type DM6008 and ZA210800

Figure 10 Complete experimental setup

by Eddy-Current Sensors The amplitude of the input sine-displacement excitation for analysis was measured as 119861 =

001mm and the damping ratio for the transverse vibrationaround the fundamental resonant frequency was found to be120577 = 00025 by Logarithmic Decrement Method

In the following paragraphs we focus on the transientand steady responses of the cantilever system with a sine baseexcitation

Figures 11 and 12 show the theoretical displacementresponses at the free end of the beam from transient tosteady state with two improved mode shapes above at thefirst resonant frequency It can be seen obviously that thedisplacement responses with two improved mode shapeswe proposed are almost identical This richly proves thatthe first-type mode shapes reformulated by the reinforcedmethod of separation of variables are equal to the second-type mode shapes derived by the developed method offorced vibration In addition it can be further obtainedthat when the frequency from the base excitation is veryclose but not exactly equal to the resonant frequency of freevibrations of the cantilever system a kind of vibration calledbeating occurs and the period of the beating increases as120596 approached 120596

1 This kind of beating decays exponentially

along the time due to the mechanical dampingFigure 13 shows the experimental displacement responses

at the position of 119909 = 1204mm from transient to steady

state As the same to the theoretical results experimental dis-placement response can be regarded as the sum of waveformsfrom transient and steady states But the transient responsevanished gradually with the time due to the mechanicaldamping and only the steady response left In addition thephenomenon of beating still existed The amplitude of thisbeating phenomenon reduced gradually and the closer theexciting frequency approached to the first natural frequencyof the cantilever system the faster the amplitude of thebeating decreased

Furthermore the amplitudes of the theoreticalwaveformsin Figures 11 and 12 are different from the experimental resultin Figure 13 This may because af the following (a) The effectof mechanical damping with respect to stiffness was onlyconsidered and the mass damping was ignored in theoreticalanalysis which can also weaken the transient vibration faster(b) The first natural frequency of the cantilever system wascalculated and measured by theoretical and experimentalmethods as 815Hz and 812Hz respectively The little dif-ference made that the max amplitudes from theoretical andexperimental results did not appear at the same frequency(c) In theoretical analysis the tip-mass was equated by theinertia force and inertia moment acting at the free end ofthe cantilever so the total length for theoretical analysiswas just the length of the beam of 1204mm Howeverduring experiments the tip-displacement was measured at

12 Shock and Vibration

005

0

minus0050 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6

0 1 2 3 4 5 6 0 1 2 3 4 5 6

0 1 2 3 4 5 6 0 1 2 3 4 5 6

0 1 2 3 4 5 6

0

0

1 2 3 4 5 6

02

01

0

minus01

minus02

02

01

0

0

minus01

minus02

015

01

005

minus005

minus01

008

006

004

002

0

minus002

minus004

minus006

minus008

008

006

004

002

0

minus002

minus004

minus006

minus008

008

006

004

002

minus002

minus004

minus006

minus008

2

15

1

05

0

minus05

minus1

minus15

minus2

01

005

0

minus005

minus01Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at735

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

795

Hz (

mm

)

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at855

Hz (

mm

)

tran

sient

to st

eady

at755

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mTh

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

815

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

875

Hz (

mm

)

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at775

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

835

Hz (

mm

)

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at895

Hz (

mm

)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Figure 11 Theoretical tip displacement responses with the second-type mode shape of the beam from transient to steady state

the approximate position of 119909 = 1204mm This is justan approximate value because Eddy-Current Sensor cannotgive exact measurement results of the displacement at thejunction between the cantilever and the tip-mass This maycause that the amplitude level from theoretical discussionwashigher than from experimental measurement In additionthe amplitude of the instantaneous base displacement fromthe exciter may be not the fixed value 001mm at the startinginstant Therefore the maximum displacement responsesduring the transient state in the experimental measurementsmay shift away from the real experimental results

The theoretical responses with two improved types ofmode shapes and the experimental displacement responsesversus varying positions of beam and the frequencies atsteady state are shown in Figures 14ndash16 respectively Figures14 and 15 show two types of theoretical displacements atthe positions of 119909 = 364mm 574mm 784mm 994mmand 1204mmwith continuous varying frequencies In exper-iments the displacement responses were measured at thesame positions with theoretical calculations but with somecertain frequencies of 782Hz 792Hz 802Hz 807Hz81Hz 812Hz 817Hz 82Hz 83Hz and 84Hz as plottedin Figure 16 It is noticed that when the frequency of baseexcitation approached to the first natural frequency of thecantilever system the amplitude of forced vibration willrapidly increase and become maximum for the case when

the frequency of the excitation coincides with the naturalfrequency Table 3 lists the values of the first natural frequen-cies and the displacement responses from theoretical andexperimental results at some certain positions long the beamunder the first natural frequencies It can be seen that thetheoretical and experimental results are matched very close

Table 3 The first natural frequencies and the resonantdisplacements at some certain positions

In a word the theoretical results with two different modeshapes derived by two improved analytical methods haveextremely the same behaviors at transient and steady stateswith respect to varying frequencies from base excitationThedisplacement responses indicate close agreement betweenthe experimental and analytical results All of these resultsgive full verification for the improvement of the analyticalmethods in this paper utilized in recalculating the modeshapes of a cantilever beam when an offset distance existsbetween the gravity center of the tip-mass and the point ofattachment

5 Conclusions

This paper presents two improved analytical methods ofcalculation for natural frequencies and mode shapes of auniform cantilever beam carrying a tip-mass under an input

Shock and Vibration 13

005

0

minus005Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at735

Hz (

mm

)

0 1 2 3 4 5 6

Time (s)

0 1 2 3 4 5 6

0 1 2 3 4 5 6

02

01

0

minus01

minus02

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at795

Hz (

mm

)

0

015

01

005

minus005

minus01

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at855

Hz (

mm

)

0 1 2 3 4 5 6

Time (s)

008

006

004

002

0

minus002

minus004

minus006

minus008tran

sient

to st

eady

at755

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

m

2

15

1

05

0

minus05

minus1

minus15

minus2Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at815

Hz (

mm

)

0 1 2 3 4 5 6

008

006

004

002

0

minus002

minus004

minus006

minus008Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at875

Hz (

mm

)

0 1 2 3 4 5 6

01

005

0

minus005

minus01Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at775

Hz (

mm

)

0 1 2 3 4 5 6

Time (s)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

0 1 2 3 4 5 6

0 1 2 3 4 5 6

0

008

006

004

002

minus002

minus004

minus006

minus008Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at895

Hz (

mm

)

02

01

0

minus01

minus02

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at835

Hz (

mm

)Figure 12 Theoretical tip displacement responses with third-type mode shape of the beam from transient to steady state

Table 3 The first natural frequencies and the resonant displacements at some certain positions

Resonant frequency or positions The values of frequencies and displacement responses (Hz or mm) ErrorsTheory based on 119880(2)

1(119909) Theory based on 119880(3)

1(119909) Experiment

The 1st natural frequency 815 815 812 037119909 = 364mm 01285 01285 01277 062119909 = 574mm 03022 03022 03001 069119909 = 784mm 05279 05279 05246 063119909 = 994mm 07905 07905 07855 063119909 = 1204mm 10742 10742 10671 066

base excitation using forced vibration theory and the methodof separation of variables A simplified cantilever model byreplacing the tip-mass with an equivalent inertial force andinertial moment acting at the free end of the cantilever beamwas detailed In addition using the Dirac Delta Functionthe base transverse excitation the concentrated equivalentinertia force and inertia moment were represented in theform of distributed loads which can make the method ofseparation of variables be utilized properly In this case sometypical results of natural frequencies and mode shapes of thecantilever beam with a tip-mass were derived using the rein-forced method of separation of variables and the developedforced vibration respectively Furthermore based on modalanalysis method the transient and steady displacement

responses of the cantilever system with a tip-mass and a sinebase excitation were presented in which two different types ofmode shapes calculated by two improved analytical methodswere used Finally an experimental cantilever system wasfabricated and some practical displacement responses weremeasured

From the results of the typical natural frequencies andmode shapes we can easily found that the mode shapesfrom the first improved type were identical to the modeshapes from the second type whether the mode shapes fromthe lower order or from the higher order even the offsetdistance was large The comparing results of the transientand steady displacement responses from the theoreticalanalysis and experimental measurements further proved the

14 Shock and Vibration

005

0

minus0050 2 4 6 8 10 12

0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12

0 2 4 6 8 10 120 2 4 6 8 10 12

0 2 4 6 8 10 12 0 2 4 6 8 10 120 2 4 6 8 10 12

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at732

Hz (

mm

)

0

008

006

004

002

minus002

minus004

minus006

minus008

0

008

006

004

002

minus002

minus004

minus006

minus008

0

008

006

004

002

minus002

minus004

minus006

minus008

0

015

01

005

minus005

minus01

01

005

0

minus005

minus01

2

15

1

05

0

minus05

minus1

minus15

minus2

03

02

01

0

minus01

minus02

03

02

01

0

minus01

minus02

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at752

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at772

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at792

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at812

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at832

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at852

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at872

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at892

Hz (

mm

)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Figure 13 Experimental displacement responses at the position of 119909 = 1204mm from transient to steady state

12

1

08

06

04

02

0120 100 80 60 40 20

78

80

82

84Theo

retic

al st

eady

disp

lace

men

t (m

m)

Position at the beam (mm)Freq

uency

(Hz)

Figure 14 Theoretical steady-displacement responses versus posi-tions and frequencies with mode 119880(2)

1(119909)

improvements of the analytical methods for calculating themode shapes of the cantilever beam with a tip-mass anda base excitation In addition the amplitudes of the modeshapes decreased with the increasing mass value of the tip-mass when the moment of inertia and offset were constantsThe amplitude of the first-order mode shape increased but

12

1

08

06

04

02

0120 100 80 60 40 20

78

80

82

84Theo

retic

al st

eady

disp

lace

men

t (m

m)

Position at the beam (mm)Freq

uency

(Hz)

Figure 15 Theoretical steady-displacement responses versus posi-tions and frequencies with mode 119880(3)

1(119909)

the other nine mode shapes decreased with the increasingmoment of inertia when the mass and offset were set asconstants Conversely the amplitude of the first mode shapedecreased but the other ninemode shapes increased with theincreasing offset distance under the condition that the massand moment of inertia were constants Globally the effect

Shock and Vibration 15

12

1

08

06

04

02

0

Expe

rimen

tal s

tead

y di

spla

cem

ent (

mm

)

120 100 80 60 40 20Position at the beam (mm)

78

80

82

84Freq

uency

(Hz)

Figure 16 Experimental steady-displacement responses versuspositions and frequencies

from varying combinations of the mass and moment inertiaof the tip-mass and the offset distance for the lower-ordermode shapes was stronger than for the higher ones

Furthermore it is clearly noticed that the mass of tip-mass can reduce the values of the natural frequencies andthe increasing mass value gave the decreasing frequencieswhen the moment of inertia of the tip-mass and the offsetdistance were set as constants Similarly the moment ofinertia can further decrease the natural frequencies and thehigher moment of inertia resulted in the lower frequencieswhen the mass and the offset were constants Differently theoffset distance can reduce the first-order natural frequenciesbut increase the other nine order frequencies and the higheroffset distance led to the lower first-order natural frequenciesbut produced the higher other nine order frequencies whenthe mass and the moment of inertia were set to constant Inglobal view the influence from the varying mass momentof inertia and offset distance for the lower-order naturalfrequencies was larger than for the higher-order ones weatherthis influence was negative or positive

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the Natural Science Foundationsof China (no 51275380) and the Scientific and TechnologicalProject of Shaanxi Province (no 2012K06-36) The authorswould like to thank the reviewers for their valuable commentson this paper

References

[1] P A A Laura J L Pombo and E A Susemihl ldquoA note on thevibrations of a clamped-free beam with a mass at the free endrdquoJournal of Sound and Vibration vol 37 no 2 pp 161ndash168 1974

[2] B Rama Bhat andHWagner ldquoNatural frequencies of a uniformcantilever with a tip mass slender in the axial directionrdquo Journalof Sound and Vibration vol 45 no 2 pp 304ndash307 1976

[3] C W S To ldquoVibration of a cantilever beam with a baseexcitation and tip massrdquo Journal of Sound and Vibration vol83 no 4 pp 445ndash460 1982

[4] P A A Laura ldquoComments on lsquoVibration of a cantilever beamwith a base excitation and tip massrsquordquo Journal of Sound andVibration vol 88 no 4 p 569 1983

[5] C W S To ldquoDiscussion on lsquovibration of a cantilever beam witha base excitation and tip massrsquordquo Journal of Sound and Vibrationvol 97 no 2 pp 349ndash351 1984

[6] R G Jacquot ldquoFurther comments on lsquoVibration of a cantileverbeam with a base excitation and tip massrsquordquo Journal of Sound andVibration vol 93 no 2 pp 312ndash313 1984

[7] S AMousavi Lajimi andG R Heppler ldquoComments on lsquonaturalfrequencies of a uniform cantilever with a tip mass slender inthe axial directionrsquordquo Journal of Sound and Vibration vol 331 no12 pp 2964ndash2968 2012

[8] R B Bhat ldquoAuthors response to the comments on lsquonaturalfrequencies of a uniform cantilever with a tip mass slender inthe axial directionrsquordquo Journal of Sound and Vibration vol 331 no12 pp 2962ndash2963 2012

[9] E Esmailzadeh and G Nakhaie-Jazar ldquoPeriodic behavior ofa cantilever beam with end mass subjected to harmonic baseexcitationrdquo International Journal of Non-Linear Mechanics vol33 no 4 pp 567ndash577 1998

[10] H Abramovich and O Hamburger ldquoVibration of a cantilevertimoshenko beam with a tip massrdquo Journal of Sound andVibration vol 148 no 1 pp 162ndash170 1991

[11] B Pratiher ldquoVibration control of a transversely excited can-tilever beam with tip massrdquo Archive of Applied Mechanics vol82 no 1 pp 31ndash42 2012

[12] M Kim M Hoegen J Dugundji and B L Wardle ldquoModelingand experimental verification of proof mass effects on vibrationenergy harvester performancerdquo Smart Materials and Structuresvol 19 Article ID 045023 2010

[13] Y T Li and R Wong ldquoIntegral and series representations ofthe Dirac delta functionrdquo Communications on Pure and AppliedAnalysis vol 7 no 2 pp 229ndash247 2008

[14] M Al-Gwaiz Sturm-LiouvilleTheory and its Applications chap-ter 3 Springer London UK 2008

[15] F Bao-lian ldquoGeneralized reciprocal theorems and their appli-cationsrdquo Applied Mathematics and Mechanics vol 23 no 2 pp203ndash210 2001

[16] C Cai H Zheng M Khan and K HungModeling of MaterialDamping Properties in ANSYS Defense Systems Division Insti-tute of High Performance Computing 2002

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2 Shock and Vibration

b(t)

y

ua(x t)

xCg

o998400

rc

EI m L

M J

Figure 1 The model of cantilever beam with a tip-mass under baseexcitation

moment and the base excitation are represented as dis-tributed loads using Dirac Delta Function In this case sometypical natural frequencies and mode shapes are calculatedby the developed and undeveloped methods The comparingresults show that the inconsistency between the mode shapesderived by the undeveloped method become larger withthe increasing offset distance However after improvementthe mode shapes derived by two developed methods are inextremely good agreement with each other even the offsetdistance between the gravity center of the tip-mass andthe attachment point is large As further verification thetransient and steady displacement responses of the cantileversystem under a sine base excitation are presented by modalanalysis method in which the mode shapes derived by twoimproved analytical methods above are separately utilizedFinally an experimental cantilever system is fabricated andthe theoretical displacement responses are validated by theexperimental measurements successfully

2 The Mathematical Model

The uniform cantilever beam carrying a tip-mass at the freeend under base excitation is shown schematically in Figure 1It should be noticed that the gravity center of the tip-massis collinear with the gravity center of beam but does notcoincide with the point of the end attachment 1199001015840 The massand the moment of inertia of the tip-mass are both taken intoaccount but the rotary inertia of the beam is neglected

21 The Simplified Model of the Cantilever Beam with a Tip-Mass According to DrsquoAlembertrsquos principle the tip-mass atthe free end of the cantilever can be equated by an inertialforce and an inertial moment while vibrating In additionthe tip-mass with larger elastic modulus is always chosencomparing to the elastic modulus of beam So during vibra-tion the deformation of tip-mass can be ignored comparingto the beam That implies that the tip-mass can be regardedas a rigid body while vibrating In this case considering thetranslational motion and rotational motion about a fixed axisof rigid body the tip-mass can be equated by an inertialforce and an inertial moment The details about these twoequivalent processing will be discussed as follows

In the following analysis we will limit our study about thetip-mass to planar kinetics because the rigid tip-mass can beconsidered to be symmetrical with respect to a fixed referenceplane

First considering the translational effect since themotion of the rigid body can be viewed within the referenceplane all the forces and couple moments acting on the bodycan then be projected into the plane All particles insidethe tip-mass may move with the same acceleration (seeFigure 2(a)) The results of the simplification of this inertialforce system to the point 1199001015840 under translational motion givean inertial force and an inertial moment as follows

F(1)119892= sum F(1)

119892119894= minussum119898

119894a119894= minussum119898

119894a119888= minus119872a

119888

M(1)119892= sum r

119894times F(1)119892119894= minussum r

119894119898119894times a119888= minus119872r

119888times a119888

(1)

where the vector a119888indicates the acceleration of the gravity

center of the tip-mass the parameter 119872 indicates the valueof the tip-mass and r

119888presents the offset distance

Next in the condition of rotational motion about an axisperpendicular to the reference plane and passing throughpoint 1199001015840 all particles inside the rigid tip-massmay rotate withthe same angular acceleration (see Figure 2(b)) Reducing theinertial rotating system to point 1199001015840 gives the other inertialforce and inertial moment as

F(2)119892= sum F(2)

119892119894= minussum119898

119894a120591119894= minussum119898

119894(120585 times r

119894) = minus119872120585 times r

119888

M(2)119892= sum r

119894times F(2)119894= minussum119898

119894(r119894times 120585 times r

119894) = minus120585sum119898

1198941199032

119894

= minus 120585 (119869 +1198721199032

119888)

(2)

where 120585 and 119869 indicate the angular acceleration and themoment of inertia of the rigid tip-mass respectively

In this case the total inertial force and inertial momentof the tip-mass can be written as the sum of the inertialequivalent forces and moments from translational and rota-tional motions Assuming that the acceleration and angularacceleration of all particles inside of tip-mass are equal tothe accelerations of the particle at point 1199001015840 then the inertialequivalent force and moment of the rigid tip-mass would bewritten in scalar quantity as

119865119892= 119872

1205972119906119886(119871 119905)

1205971199052

+ 119903119888119872

1205972

1205971199052[

120597119906119886(119871 119905)

120597119909

]

119872119892= 119903119888119872

1205972119906119886(119871 119905)

1205971199052

+ (119869 + 1199032

119888119872)

1205972

1205971199052[

120597119906119886(119871 119905)

120597119909

]

(3)

where 119906119886(119871 119905) indicates the absolute transverse displacement

at the free end of the cantilever beam The directions of thewhole inertial equivalent force and moment can be foundaccording to Newtonrsquos third law of motion as shown inFigure 3

22 The Reinforced Method of Separation of Variables Inthis section the natural frequencies and mode shapes of thecantilever model (as shown in Figure 3) will be derived bythe reinforced method of separation of variables First usingDiracDelta Function [13] the concentrated equivalent inertia

Shock and Vibration 3

ri

rcCg

mi

aiac

F(1)gi

F(1)g

o998400

M(1)g

(a)

ri

rcCg

mi

acF(2)gi

F(2)g

o998400

a120591i

M(2)g

120585

(b)

Figure 2 Simplification of the tip-mass to point 1199001015840 under (a) translational motion and (b) rotational motion

y

x

b(t) ua(x t)

FgMg

EI m L

Figure 3 The simplified model of cantilever with tip-mass beingequated by inertial force and moment

force 119865119892and inertia moment 119872

119892can be represented in the

form of distributed loads as

119891119886(119909 119905)

= 119865119892120575 (119909 minus 119871)

= [119872

1205972119906119886(119909 119905)

1205971199052

+ 119903119888119872

1205972

1205971199052(

120597119906119886(119909 119905)

120597119909

)] 120575 (119909 minus 119871)

119872119886(119909 119905)

= 119872119892120575 (119909 minus 119871)

= [119903119888119872

1205972119906119886(119909 119905)

1205971199052

+ (119869 + 1199032

119888119872)

1205972

1205971199052(

120597119906119886(119909 119905)

120597119909

)]

times

119889120575 (119909 minus 119871)

119889119909

(4)

Therefore employing variational method the dynamicmotion equation of the simplified model of the cantileverbeam with a tip-mass under base transverse excitation isobtained as

119864119868

1205974119906119886(119909 119905)

120597119909

+ 119898

1205972119906119886(119909 119905)

120597119905

= 119891119886(119909 119905) minus 119872

119886(119909 119905) (5)

where 119864119868 and119898 denote the flexural rigidity and the mass perunit length of the beam respectively

The absolute displacement can be considered as thesum of the relative displacement and the base displacementexcitation

119906119886(119909 119905) = 119906rel (119909 119905) + 119887 (119905) (6)

Substituting (6) into (5) gives the new form of the dynamicmotion equation as

119864119868

1205974119906rel (119909 119905)

120597119909

+ 119898

1205972119906rel (119909 119905)

120597119905

= [119872

1205972119906rel (119909 119905)

1205971199052

+ 119903119888119872

1205972

1205971199052(

120597119906rel (119909 119905)

120597119909

)] 120575 (119909 minus 119871)

+ [119903119888119872

1205972119906rel (119909 119905)

1205971199052

+ (119869 + 1199032

119888119872)

1205972

1205971199052(

120597119906rel (119909 119905)

120597119909

)]

times

119889120575 (119909 minus 119871)

119889119909

+ [119872120575 (119909 minus 119871) + 119903119888119872

119889120575 (119909 minus 119871)

119889119909

minus 119898]

1198892119887 (119905)

1198891199052

(7)

In this case the simplified model of the cantilever beamunder base excitation can be considered as a new analyticalmodel with the dynamic differential equation of (8) and theboundary conditions of (9) through (12)

119864119868

1205974119906119886(119909 119905)

120597119909

+ 119898

1205972119906119886(119909 119905)

120597119905

= [119872120575 (119909 minus 119871) + 119903119888119872

119889120575 (119909 minus 119871)

119889119909

minus 119898]

1198892119887 (119905)

1198891199052

(8)

at 119909 = 0

119906119886(119909 119905) = 0 (9)

119889

119889119909

119906119886(119909 119905) = 0 (10)

4 Shock and Vibration

at 119909 = 119871

119864119868

1198893

1198891199093119906119886(119909 119905) = [119872

1205972119906119886(119909 119905)

1205971199052

+ 119903119888119872

1205972

1205971199052(

120597119906119886(119909 119905)

120597119909

)]

(11)

119864119868

1198892

1198891199092119906119886(119909 119905)

= minus [119903119888119872

1205972119906119886(119909 119905)

1205971199052

+ (119869 + 1199032

119888119872)

1205972

1205971199052(

120597119906119886(119909 119905)

120597119909

)]

(12)

The solutions of (8) subjected to four boundary condi-tions and two initial conditions can be obtained convenientlyby the traditional method of separation of variables Themethod regards the response as a superposition of the systemeigen-functionsmultiplied by corresponding time dependentgeneralized coordinates Of course this necessitates firstobtaining the solution of the system eigen-value problemTurning attentions to the corresponding eigen-value problemand considering the free vibration characterized by setting119887(119905) = 0 the solution of (8) becomes separable in space andtime Letting

119906119886(119909 119905) = 119880 (119909) 119879 (119905) (13)

and substituting (13) into the free vibration form of (8) twodependent equations are obtained using the separation ofvariables method as

119889

1198891199052119879 (119905) + 120596

2119879 (119905) = 0 (14)

1198894

1198891199094119880 (119909) minus 120573

4119880 (119909) = 0 (15)

where 1205962 indicates the angular frequency and 1205734 =

1198981205962(119864119868) In this case the four boundary conditions will be

changed as

at 119909 = 0

119880 (119909) = 0 (16)

119889

119889119909

119880 (119909) = 0 (17)

at 119909 = 119871

119864119868

1198893

1198891199093119880 (119909) = minus [120596

2119872119880(119909) + 119903

1198881198721205962(

119889

119889119909

119880 (119909))] (18)

119864119868

1198892

1198891199092119880 (119909) = 119903

1198881198721205962119880 (119909) + (119869 + 119903

2

119888119872)1205962(

119889

119889119909

119880 (119909))

(19)

The general solution of (15) can be easily derived to be

119880 (119909) = 1198621cos (120573119909) + 119862

2sin (120573119909) + 119862

3cosh (120573119909)

+ 1198624sinh (120573119909)

(20)

where 119862119899(119899 = 1 2 3 4) are constants of integration In

order to evaluate three of these constants in terms of thefourth and derive the characteristic equation the boundaryconditions (16) to (19) must be used Indeed substituting (20)into (16) through (19) gives the characteristic equation andthe mode shapes as

1 + 119888119888ℎ minus 120601119906 (119904119888ℎ minus 119888119904ℎ)

minus 1198771[119904119888ℎ + 119888119904ℎ minus 120601119906 (1 minus 119888119888ℎ)]

minus 21198772119904119904ℎ minus 119877

2

2(1 minus 119888119888ℎ) = 0

(21)

119880 (119909) = 1198622sin (120573119909) minus 119904ℎ (120573119909) minus 120582 [cos (120573119909) minus 119888ℎ (120573119909)]

(22)

where 120582 = 1198601211986011and the constant 119862

2is an arbitrary value

and in which

11986011= minus(1 + 119877

2)119888 + (119877

2minus 1)119888ℎ + 119877

1(119904 + 119904ℎ) 119860

12=

1198771(119888ℎ minus 119888) minus (1 + 119877

2)119904 + (119877

2minus 1)119904ℎ

119906 = 120573119871 119904 = sin(119906) 119888 = cos(119906) 119904ℎ = sinh(119906) 119888ℎ =cosh(119906)

120601 = 119872(119898119871) 1198771= 120601119906

3[(119870119871)

2+ (119903119888119871)2] 1198772=

1206011199062(119903119888119871) 119870 = radic119869119872

23TheDevelopedMethod of Forced Vibration If we treat thebase displacement excitation as a kind of boundary conditionthen the differential equation of motion and four boundaryconditions of the cantilever system can be written as

119864119868

1205974119906119886(119909 119905)

120597119909

+ 119898

1205972119906119886(119909 119905)

120597119905

= 0 (23)

at 119909 = 0

119906119886(0 119905) = 119887 (119905) (24)

119889

119889119909

119906119886(0 119905) = 0 (25)

at 119909 = 119871

119864119868

1198893

1198891199093119906119886(119909 119905)

= [119872

1205972119906119886(119909 119905)

1205971199052

+ 119903119888119872

1205972

1205971199052(

120597119906119886(119909 119905)

120597119909

)]

(26)

119864119868

1198892

1198891199092119906119886(119909 119905)

= minus [119903119888119872

1205972119906119886(119909 119905)

1205971199052

+ (119869 + 1199032

119888119872)

1205972

1205971199052(

120597119906119886(119909 119905)

120597119909

)]

(27)

Shock and Vibration 5

Under the simple harmonic excitation 119890119895120596119905 the steady statesolution of (23) can be regarded with the form as

119906119886(119909 119905) = 119880

119886(119909 120596) 119890

119895120596119905minus120593

= [1198611cos (120573119909) + 119861

2sin (120573119909)] 119890119895120596119905minus120593

+1198613cosh (120573119909) + 119861

4sinh (120573119909)]

(28)

where 119861119899(119899 = 1 2 3 4) are arbitrary constants the

parameter 120573 is defined by 1205734 = 1198981205962(119864119868) 119895 is the imaginaryunit and 120593 is the lagging phase

We recalculated the natural frequencies and the modeshapes directly by substituting (28) into four boundaryconditions (24) through (27) and obtained the characteristicequation and displacement response which are different from[3]

119863 = 1 + 119888119888ℎ minus 120601119906 (119904119888ℎ minus 119888119904ℎ)

minus 1198771[119904119888ℎ + 119888119904ℎ minus 120601119906 (1 minus 119888119888ℎ)]

minus 21198772119904119904ℎ minus 119877

2

2(1 minus 119888119888ℎ)

119880 (119909) = (

119861

2119863

) 1198651015840

1119906cos (120573119909) + 1198651015840

2119906cosh (120573119909)

+1198651015840

3119906[sin (120573119909) minus sinh (120573119909)]

(29)

where 119861 is defined by 119887(119905) = 119861119890119895120596119905 and in which

1198651015840

1119906= 1+119888119888ℎminus119904119904ℎminus2120593119906119904119888ℎminus2119877

1119888119904ℎ+(119877

1120593119906minus119877

2

2)(1minus

119888119888ℎ + 119904119904ℎ) minus 21198772(119888119888ℎ + 119904119904ℎ) =119865

1119906minus 21198772(119888119888ℎ + 119904119904ℎ)

1198651015840

2119906= 1+119888119888ℎ+119904119904ℎ+2120593119906119888119904ℎminus2119877

1119904119888ℎ+(119877

1120593119906minus119877

2

2)(1minus

119888119888ℎ minus 119904119904ℎ) + 21198772(119888119888ℎ minus 119904119904ℎ) =119865

2119906+ 21198772(119888119888ℎ minus 119904119904ℎ)

1198651015840

3119906= (119904119888ℎ + 119888119904ℎ)(1 minus 119877

1120593119906 +119877

2

2) + 2[120593119906119888119888ℎ minus 119877

1119904119904ℎ minus

1198772(119904119888ℎ minus 119888119904ℎ)] = 119865

3119906

It is obvious that the characteristic equation derived by thereinforced method of separation of variables is equal tothe equation obtained by the developed method of forcedvibration

Themode shapes cannot be obtained directly because theboundary conditions contain the parameter1205962 However thisproblem can be successfully solved by Dirac Delta Function[13] Thus the mode shapes may be obtained by the Sturm-Liouville theory [14] and expressed as

119880119894(119909) = (

1

2

) 1198651015840

1119906119894119865 cos (120573

119894119909) + 119865

1015840

2119906119894cosh (120573

119894119909)

+ 1198651015840

3119906119894[sin (120573

119894119909) minus sinh (120573

119894119909)]

(30)

3 Computed Results and Comparisons

In this section three types of the natural mode shapes ofthe cantilever beam with a tip-mass were computed and

compared The first one is called the undeveloped modeshapes which is directly obtained from [3] by the expression

119880(1)

119894(119909) = (

1

2

) 1198651119894119888

cos(119906119894119909

119871

) + 1198652119894119888

cosh (119906119894119909

119871

)

+ 1198653119894119888[sin(

119906119894119909

119871

) minus sinh(119906119894119909

119871

)]

(31)

where 1198651119894119888

= 1198651119894119888ℎ(119906119894) 1198652119894119888

= 1198652119894119888ℎ(119906119894) and 119865

3119894119888=

1198653119894119888ℎ(119906119894) in which 119865

1119894 1198652119894 and 119865

3119894are the functions 119865

1119906 1198652119906

and 1198653119906

with the variable 119906 replaced by 119906119894 respectively

The second one is based on the expression of thedeveloped mode shapes derived in this paper Let (30) bemultiplied by 21198651015840

3119894to give the second mode shapes as

119880(2)

119894(119909) = sin(

119906119894119909

119871

) minus sinh(119906119894119909

119871

) + (

1198651015840

1119894

1198651015840

3119894

) cos(119906119894119909

119871

)

+ (

1198651015840

2119894

1198651015840

3119894

) cosh (119906119894119909

119871

)

(32)

At last the arbitrary constant 1198622in (22) was chosen as

constant 1 to give the third type of the natural mode shapes as

119880(3)

119894(119909) = sin(

119906119894119909

119871

) minus 119904ℎ (

119906119894119909

119871

)

minus 120582 [cos(119906119894119909

119871

) minus 119888ℎ (

119906119894119909

119871

)]

(33)

Before the mode shapes are determined the roots ofthe characteristic equation must be calculated firstly Thecharacteristic equation (21) or 119863 = 0 can be solved by twosteps First approximate roots were pointed out by applyinggraphing method Then the approximate roots were iteratedby the bisection method to give the exact characteristic rootswith given error All these processes were finished usingMATLAB software

Because the condition that the gravity center of the tip-mass coincided with the point of the end attachment hasbeen verified in [6] we focus on the condition when theoffset distance is not zero Table 1 shows the first five roots ofthe characteristic equations from five samples with differentvalues of 120601 119896119871 and 119889119871 It can be clearly noticed thatthe larger value of 119889119871 resulted in the less value of the firstcharacteristic root but the larger values of the other fourcharacteristic roots when 120601 and 119896119871 were set as constants

Once the roots of the characteristic equation have beenobtained three types of the first five mode shapes of thecantilever structure from the five given samples can becalculated Figures 4 5 6 7 and 8 show the first five modeshapes of the cantilever with various combinations of 120601 119896119871and 119889119871

6 Shock and Vibration

Table 1 Typical roots of characteristic equation with variables 120601 119896119871 and 119889119871

Sample 120601 119870119871 119889119871

Roots of characteristic equation1199061

1199062

1199063

1199064

1199065

1 4 4 0 035211 129234 477818 788401 11017802 4 4 05 034814 130637 477952 788472 11018273 4 4 1 034200 132825 478225 788634 11019404 4 4 2 032544 138983 479177 789233 11023685 4 4 4 028806 154543 482543 791455 1103990

003002001

0

0 05 1 0 05 1 0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

000000

050000

100000

i = 1

000000

050000

100000

i = 1

000000 000000

000000

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

000000

050000

100000

i = 2

000000

050000

100000

i = 2

minus186571 times 10minus9

000792

003152

minus198560 times 10minus8

minus9418658

minus18545378

0

minus100

minus2000 05 1

0 05 1

0 05 1

0 05 1

1050

times104 i = 3

000000

050000

100000

i = 3

000000

050000

100000

i = 3

786191 times 10minus6

111026 times 104

minus686262 times 103

50

minus5

times105 i = 4000000

050000

100000

i = 4

000000

050000

100000

i = 4

i = 5000000

050000

100000

i = 5

000000 000000

050000

100000

i = 5times106

20

minus2

000017

minus271300 times 106

minus855944 times 104

1

05

0

1

05

0

03

02

01

0

03

02

01

0

151

050

151

050

2

0

minus2

2

0

minus2

210

minus1

210

minus1

minus493414 times 10minus8

032943

131050

367824 times 10minus11

017448

034354

113682 times 10minus10

160541

minus009923

457130 times 10minus11

minus002114

006144

866744 times 10minus11

minus140614

minus004436

032943

131050

017448

034354

160541

minus009923

000000

minus002114

006144

minus140614

minus004436

Notes at xL Notes at xL Notes at xLValues ValuesValues

Ui(x)

xL

Ui(x)

xL

Ui(x)

xL

minus233509 times 10minus5

107963 times 104

minus313859 times 104

First-type mode shapes U(1)i

(x) Second-type mode shapes U(2)i

(x) Third-type mode shapes U(3)i

(x)

Figure 4 Three types of the first five mode shapes with the combinations of 120601 = 4 119896119871 = 4 and 119889119871 = 0

It is obvious that when 119889119871 = 0 the first type of naturemode shapes 119880(1)

119894(119909) derived by the undeveloped method

turned far away from the third-type mode shapes 119880(3)119894(119909)

derived by the reinforced method of separation of variablesIn addition the larger value of 119889119871 resulted in the moreinconsistent mode shapes However the second type of modeshapes 119880(2)

119894(119909) obtained by the developed method of forced

vibration seemed in good agreement with the third onesFurthermore the amplitude of the first mode shape

decreased but the other four mode shapes increased with theincreasing value of the 119889119871 when 119896119871 and 120601 were fixed to be

constants The effect from the varying value of 119889119871 for thelower mode shapes was always stronger than for the higher

4 Experimental Verification

In order to experimentally explain and compare two typesof natural mode shapes derived by two improved methodsabove the transient and steady responses of the cantileversystem under an input base excitation should be calculatedfirst This whole calculation can be finished by method ofmodal analysis But above all the orthogonality property of

Shock and Vibration 7

0 05 1

0 05 1

Ui(x)

xL

Ui(x)

xL

Ui(x)

xL

014013012011

0

minus100

minus200

000000

050000

100000

i = 1

000000

050000

100000

i = 2

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

011044

011840

014143

533123

minus9287788

minus18666064

0 05 1

1050

times104

000000

050000

100000

i = 3minus8449356

107519 times 105

minus970528 times 103

0 05 1

50

minus5

times105

000000

050000

100000

i = 422493344

139452 times 104

139108 times 105

0 05 1

40

minus4minus8

times106

000000

050000

100000

i = 5minus42806116

minus266045 times 106

minus765307 times 106

0 05 1

000000

050000

100000

i = 106

04

02

0

392120 times 10minus9

017490

069258

0 05 1

06

04

02

0

0 05 1

000000

050000

100000

i = 2547973 times 10minus11

017881

034942

03

02

01

00 05 1

03

02

01

0

0 05 1

000000

050000

100000

i = 3minus505260 times 10minus11

160517

minus010133

151

050

0 05 1

151

050

0 05 1

000000

050000

100000

i = 4minus776048 times 10minus11

minus002162

006261

2

0

minus20 05 1

2

0

minus2

0 05 1

000000

050000

100000

i = 5663249 times 10minus11

minus140614

minus004516

210

minus1

000000

000000

000000

000000

017490

069258

017881

034942

160517

minus010133

minus002162

006261

000000

050000

100000

i = 5

0 05 1

210

minus1

000000

minus140614

minus004516

Notes at xL Notes at xL Notes at xLValues ValuesValuesFirst-type mode shapes U(1)i

(x) Second-type mode shapes U(2)i

(x) Third-type mode shapes U(3)i

(x)

Figure 5 Three types of the first five mode shapes with the combinations of 120601 = 4 119896119871 = 4 and 119889119871 = 05

034

033

032

031

0 05 1 0 05 1 0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1 0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0

minus100

minus200

10

5

0

times104

50

minus5

times105

times106

0minus5minus10minus15

Ui(x)

xL

Ui(x)

xL

Ui(x)

xL

04

02

0

04

02

0

04

02

0

04

02

0

151

050

151

050

2

0

minus2

2

0

minus2

210

minus1

210

minus1

000000

050000

100000

i = 1

000000 000000

050000

100000

i = 1

030899

031712

033934

000000

050000

100000

i = 2861965

minus9105604

minus18440224

000000

050000

100000

i = 3minus10637851

973291 times 104

minus123754 times 104

000000

050000

100000

i = 419388076

168672 times 104

311790 times 105

000000

050000

100000

i = 5minus19120922

minus244086 times 106

minus152554 times 107

000000

050000

100000

i = 1

331605 times 10minus9

012125

047836

000000

050000

100000

i = 2

minus550109 times 10minus11

018658

036219

000000

050000

100000

i = 3

minus175471 times 10minus11

160459

minus010631

000000

050000

100000

i = 4

109777 times 10minus10

minus002275

006562

000000

050000

100000

i = 5

836395 times 10minus11

minus140614

minus004731

012125

047836

000000 000000

050000

100000

i = 2

018658

036219

000000 000000

050000

100000

i = 4

minus002275

006562

000000 000000

050000

100000

i = 5

minus140614

minus004731

000000 000000

050000

100000

i = 3

160459

minus010631

Notes at xL Notes at xL Notes at xLValues ValuesValuesFirst-type mode shapes U(1)i

(x) Second-type mode shapes U(2)i

(x) Third-type mode shapes U(3)i

(x)

Figure 6 Three types of the first five mode shapes with the combinations of 120601 = 4 119896119871 = 4 and 119889119871 = 1

8 Shock and Vibration

088087086085

0 05 1 0 05 1 0 05 1

0 05 1 0 05 1

0 05 10 05 1

0 05 1 0 05 1

0 05 1 0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0

minus50

minus100

minus150

6420

minus2

times104

840

minus4

times105

0minus1minus2minus3

times107

0302010

03

02

01

0

04

02

0

04

02

0

151

050

151

050

210

minus1

210

minus1

210

minus1

210

minus1

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

084906

085736

087731

minus1308377

minus9308045

minus16072843

63490847

565642 times 104

minus172997 times 104

minus329762092

230021 times 104

667852 times 105

946242416

minus149912 times 106

minus306922 times 107

minus995048 times 10minus10

007811

030669

511030 times 10minus11

021118

040600

minus114494 times 10minus10

160244

minus012466

688252 times 10minus11

minus002686

007708

183045 times 10minus11

minus140613

minus005561

000000

000000

000000

000000

000000

007811

030669

021118

040600

160244

minus012466

minus002686

007708

minus140613

minus005561

Ui(x)

xL

Ui(x)

xL

Ui(x)

xL

Notes at xL Notes at xL Notes at xLValues ValuesValuesFirst-type mode shapes U(1)i

(x) Second-type mode shapes U(2)i

(x) Third-type mode shapes U(3)i

(x)

Figure 7 Three types of the first five mode shapes with the combinations of 120601 = 4 119896119871 = 4 and 119889119871 = 2

the natural mode shapes should be derived In fact usingBetti theorem [15] the orthogonality properties with respectto mass and stiffness can be obtained respectively as follows

int

119871

0

119898119880(119896)

119903(119909)119880(119896)

119894(119909) 119889119909 +119872119880

(119896)

119903(119871)119880(119896)

119894(119871)

+ 119903119888119872119880(119896)

119894(119871)

119889119880(119896)

119903(119871)

119889119909

+ 119903119888119872119880(119896)

119903(119871)

119889119880(119896)

119894(119871)

119889119909

+ (119869 + 1199032

119888119872)

119889119880(119896)

119894(119871)

119889119909

119889119880(119896)

119903(119871)

119889119909

= (120575(119896)

119903119894)

2

(119896 = 1 2 119903 119894 = 1 2 )

(34)

int

119871

0

119864119868

1198892119880(119896)

119894(119909)

1198891199092

1198892119880(119896)

119903(119909)

1198891199092

119889119909 = (120596119903120575(119896)

119903119894)

2

(119896 = 1 2 119903 119894 = 1 2 )

(35)

In this case the normalized mode shapes can be defined as

119880(119896)

119903119873(119909) =

119880(119896)

119903(119909)

120575(119896)

119903119903

(36)

where

(120575(119896)

119903119903)

2

= int

119871

0

119898[119880(119896)

119903(119909)]

2

119889119909 +119872[119880(119896)

119903(119871)]

2

+ 2119903119888119872119880(119896)

119903(119871) [

119889

119889119909

119880(119896)

119903(119871)]

+ (119869 +1198721199032

119888) [

119889

119889119909

119880(119896)

119903(119871)]

2

(37)

41 Theoretical Responses of the Cantilever System with a BaseSine Excitation According to the method of modal analysisthe solution of (8) has the form of

119906119886(119909 119905) =

infin

sum

119903=1

119880(119896)

119903119873(119909) 119902119903(119905) (38)

So inserting (38) into (8) gives

119864119868

infin

sum

119903=1

1198894119880(119896)

119903119873(119909)

1198891199094

119902119903(119905) + 119898

infin

sum

119903=1

119880(119896)

119903119873(119909)

1198892119902119903(119905)

1198891199052

= [119872120575 (119909 minus 119871) + 119903119888119872

119889120575 (119909 minus 119871)

119889119909

minus 119898]

1198892119887 (119905)

1198891199052

(39)

Considering the orthogonality properties of equations (34)and (35) let (39) be multiplied by119880(119896)

119894119873(119909) and integrated over

Shock and Vibration 9

192

191

19

0 05 1 0 05 1

0 05 1 0 05 1

0 05 1

0 05 10 05 1

0 05 10 05 1

0 05 1 0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0

minus200

minus400

minus600

0minus5minus10

times104

1

0

minus1

times106

times107

0minus2minus4minus6

02

01

0

02

01

0

06

04

02

0

06

04

02

0

15

1

05

0

15

1

05

0

210

minus1

210

minus1

210

minus1

210

minus1

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

000000

050000

100000

i = 1

000000

050000

100000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

189913

190621

192051

minus56420429

minus28138349

4723398

134513 times 104

minus110859 times 105

minus260952 times 104

minus615688 times 104

496921 times 104

145091 times 106

168417 times 105

241806 times 106

minus633899 times 107

137503 times 10minus9

004964

019458

minus238241 times 10minus11

028530

054402

159443

minus019150

minus283947 times 10minus11

minus004212

012050

771494 times 10minus11

minus140605

minus008755

000000

000000

000000 000000

000000

000000

004964

019458

028530

054402

159443

minus019150

minus004212

012050

minus140605

minus008755

Ui(x)

xL

Ui(x)

xL

Ui(x)

xL

Notes at xL Notes at xL Notes at xLValues ValuesValues

minus387890 times 10minus11

First-type mode shapes U(1)i

(x) Second-type mode shapes U(2)i

(x) Third-type mode shapes U(3)i

(x)

Figure 8 Three types of the first five mode shapes with the combinations of 120601 = 4 119896119871 = 4 and 119889119871 = 4

the domain 0 lt 119909 lt 119871 to give a set of independent ordinarydifferential equations as

1198892119902119894(119905)

1198891199052

+ 1205962

119894119902119894(119905) = 119891

(119896)

119894(119905) (40)

Considering the properties of Dirac Delta Function

int

119871

0

119880(119896)

119894119873(119909) 120575 (119909 minus 119871) 119889119909 = 119880

(119896)

119894119873(119871)

int

119871

0

119880(119896)

119894119873(119909)

119889120575 (119909 minus 119871)

119889119909

119889119909 = minus

119889119880(119896)

119894119873(119871)

119889119909

(41)

the generalized forces associated with the generalized coordi-nates 119902

119894(119905) can be obtained as

119891(119896)

119894(119905)

= [119872119880(119896)

119894119873(119871) minus 119903

119888119872

119889119880(119896)

119894119873(119871)

119889119909

minus 119898int

119871

0

119880(119896)

119894119873(119909) 119889119909]

1198892119887 (119905)

1198891199052

(42)

Furthermore considering the mechanical damping asRayleigh damping and ignoring the part of mass dampingthe damping coefficient 120577 of the bimorph beam can be deter-mined by the modal damping ratio which can be obtained

by experiment [16] The independent ordinary differentialequations with damping then can be written as

1198892119902119894(119905)

1198891199052

+ 2120596119894120577

119889119902119894(119905)

119889119905

+ 1205962

119894119902119894(119905)

= [119872119880(119896)

119894119873(119871) minus 119903

119888119872

119889119880(119896)

119894119873(119871)

119889119909

minus 119898int

119871

0

119880(119896)

119894119873(119909) 119889119909]

1198892119887 (119905)

1198891199052

(43)

Now (43) can be solved using Laplace transform Applyinga sine excitation 119887(119905) = 119861 sin(120596119905) to the base of cantileverstructure under zero initial conditions the transfer functionof the independent differential equations for the cantileverbeam with tip-mass under base displacement excitation canbe written as

119866119894(119904)

=

119876119894(119904)

119861119894(119904)

=

(minus1205962)[119872119880

(119896)

119894119873(119871)minus119903

119888119872(119889119880

(119896)

119894119873(119871) 119889119909)minus119898int

119871

0119880(119896)

119894119873(119909) 119889119909]

1199042+ 2120596119894120577119904 + 120596

2

119894

(44)

10 Shock and Vibration

where 119876119894(119904) and 119861

119894(119904) are the Laplace transforms of 119902

119894(119905) and

119887(119905) respectively The inverse transformation of the transferfunction will be further obtained as

119892119894(119905)

=

(minus1205962)[119872119880

(119896)

119894119873(119871)minus119903

119888119872(119889119880

(119896)

119894119873(119871) 119889119909)minus119898int

119871

0119880(119896)

119894119873(119909) 119889119909]

120596119894radic1 minus 120577

2

times 119890minus120596119894120577119905 sin(120596

119894radic1 minus 120577

2119905)

(45)

In this case the solutions of (43) can be solved by convolutiontheorem as

119902119894(119905) = 119887 (119905) lowast 119892

119894(119905) = int

119905

0

119887 (120591) 119892119894(119905 minus 120591) 119889120591 (46)

Finally substituting the result of (46) into (38) and after somelengthy algebraic manipulation the absolute displacementresponse of the cantilever beam with tip-mass under basetransverse excitation can be obtained as

119906119886(119909 120596 119905) =

infin

sum

119894=1

119880(119896)

119894119873(119909)

119861 [119872119880(119896)

119894119873(119871) minus 119903

119888119872(119889119880

(119896)

119894119873(119871) 119889119909) minus 119898int

119871

0119880(119896)

119894119873(119909) 119889119909]

radic1205964+ 2 (2120577

2minus 1) 120596

21205962

119894+ 1205964

119894

times [

1205963

120596119894radic1 minus 120577

2

119890minus120596119894120577119905 sin(120596

119894radic1 minus 120577

2119905 + 1206011) + 1205962 sin (120596119905 + 120601

2)]

(47)

where 1206011and 120601

2are determined by tan(120601

1) = 2120577radic1 minus 120577

2

((120596120596119894)2+ 21205772minus 1) and tan(120601

2) = minus2120577(120596120596

119894)(1 minus (120596120596

119894)2)

It can be seen that the solution consists of two parts abouttimeThe first term represents free vibration which will decaywith time exponentially The second term depending on theexcitation represents the forced vibration of the system

42 Experimental Responses and Comparisons with the Theo-retical Results In order to explain the absolute displacementresponse of the cantilever structure under base excitationexperimentally a typical cantilever beam in which theelongated beam was inserted and fixed into the tip-mass tomake sure the gravity center of tip-mass was collinear withthe beam was fabricated (as shown in Figure 9) The chosengeometric and physical properties of the cantilever beamincluding the tip-mass are given in Table 2

The complete experimental devices are shown inFigure 10 The electromagnetic vibrating table type SW-TFAcontrolled by a programmable controller type SW-F-2TF-2was used to create a specific harmonic input excitationThe cantilever beam with a tip-mass was clamped onthe vibrating top platform Eddy-Current Sensors typeDM6008 and ZA210800 were used to measure the inputdisplacement from base vibrating exciter and the absolutedynamic displacement from the beam respectively Allmeasurements from cantilever beam and vibrating exciterwere connected to Digital Oscilloscope type DS1204Band the Industrial Computer including a PC-based DataAcquisitionModule from ADLINK company type DAQ2213The Digital Oscilloscope and Industrial Computer displayedand processed the measurement results using the DAQPilotsoftware and the results from subsequent analysis using theMATLAB software

Table 2 Characteristic properties of the cantilever beam includingtip-mass

Physical properties Beam Tip-massYoungrsquos modulus 119864 (GPa) 69 mdashDensity 120588

119904or 120588119898(kgm3) 2630 7753

Length 119871 or 119871119898(mm) 1204 156

Thickness 119905119904or 119905119898(mm) 49 295

Width 119908 (mm) 102 102Total mass119872 (g)a mdash 3383Offset 119903

119888(mm)b mdash 801

Rotary inertia 119869 (kgsdotm2)b c mdash 333 times 10minus6aThe total mass of the tip-mass including the elongated part of beam wascalculated by119872 = 119871

119898119905119898119908120588119898+ 119871119904119905119904119908(120588119904minus 120588119898) where 119871

119904= 10mm

bThe offset distance and the rotary inertia were calculated according to thegeometry and material properties of the beam and tip-mass using equations119903119888= (120588

1198981199051198981198712

119898minus 1205881198981199051199041198712

119904+ 1205881199041199051199041198712

119904)(2[120588

119898119905119898119871119898+ 119871119904119905119904(120588119904minus 120588119898)])

119869 = 120588119898119871119898119905119898119908 [(112)(119871

2

119898+ 1199052

119898) + ((12)119871

119898minus 119903119888)2] + 119871

119904119905119904119908(120588119904minus

120588119898)[(112)(119871

2

119904+ 1199052

119904) + (119903119888minus (12)119871

119904)2]

cThe corresponding parameters 120601 119896119871 and 119903119888119871 were calculated as follows

120601 = 119872(120588119904119905119904119908) = 21377 119896119871 = radic119869119872119871 = 00824 and 119903

119888119871 = 00665

The analytical model defined by (47) in which two typesof mode shapes derived by two improved analytical methodsare utilized can be used to analyze the steady and transientdisplacement responses of the cantilever beam with a tip-mass and an input base excitation based on the chosenvarying frequencies time and positions along the lengthof the beam Typically the displacement responses frompositions along 119909-axis at 119909

1= 364mm 119909

2= 574mm

1199093

= 784mm 1199094

= 994mm and 1199095

= 1204mmoff the origin point (as shown in Figure 9) were measured

Shock and Vibration 11

y

b(t)

x1

x2

x3

x4

x5

Ls

L

t s

t m x

Lm

rc

Cg

o998400

Figure 9 Typical cantilever beam with a tip-mass

1 2

3

56

7

8

4

Tip-massCantilever beam

(7) Cantilever with a tip-mass

(1) Industrial computer including data acquisitionmodule type DAQ2213

(2) Digital oscilloscope type DS1204B(3) Programmable controller type SW-F-FTF-2(4) Power supply type DP1308A(5) Driver modules for Eddy-current sensors(6) Electromagnetic vibrating table type SW-TFA

(8) Eddy-current sensors type DM6008 and ZA210800

Figure 10 Complete experimental setup

by Eddy-Current Sensors The amplitude of the input sine-displacement excitation for analysis was measured as 119861 =

001mm and the damping ratio for the transverse vibrationaround the fundamental resonant frequency was found to be120577 = 00025 by Logarithmic Decrement Method

In the following paragraphs we focus on the transientand steady responses of the cantilever system with a sine baseexcitation

Figures 11 and 12 show the theoretical displacementresponses at the free end of the beam from transient tosteady state with two improved mode shapes above at thefirst resonant frequency It can be seen obviously that thedisplacement responses with two improved mode shapeswe proposed are almost identical This richly proves thatthe first-type mode shapes reformulated by the reinforcedmethod of separation of variables are equal to the second-type mode shapes derived by the developed method offorced vibration In addition it can be further obtainedthat when the frequency from the base excitation is veryclose but not exactly equal to the resonant frequency of freevibrations of the cantilever system a kind of vibration calledbeating occurs and the period of the beating increases as120596 approached 120596

1 This kind of beating decays exponentially

along the time due to the mechanical dampingFigure 13 shows the experimental displacement responses

at the position of 119909 = 1204mm from transient to steady

state As the same to the theoretical results experimental dis-placement response can be regarded as the sum of waveformsfrom transient and steady states But the transient responsevanished gradually with the time due to the mechanicaldamping and only the steady response left In addition thephenomenon of beating still existed The amplitude of thisbeating phenomenon reduced gradually and the closer theexciting frequency approached to the first natural frequencyof the cantilever system the faster the amplitude of thebeating decreased

Furthermore the amplitudes of the theoreticalwaveformsin Figures 11 and 12 are different from the experimental resultin Figure 13 This may because af the following (a) The effectof mechanical damping with respect to stiffness was onlyconsidered and the mass damping was ignored in theoreticalanalysis which can also weaken the transient vibration faster(b) The first natural frequency of the cantilever system wascalculated and measured by theoretical and experimentalmethods as 815Hz and 812Hz respectively The little dif-ference made that the max amplitudes from theoretical andexperimental results did not appear at the same frequency(c) In theoretical analysis the tip-mass was equated by theinertia force and inertia moment acting at the free end ofthe cantilever so the total length for theoretical analysiswas just the length of the beam of 1204mm Howeverduring experiments the tip-displacement was measured at

12 Shock and Vibration

005

0

minus0050 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6

0 1 2 3 4 5 6 0 1 2 3 4 5 6

0 1 2 3 4 5 6 0 1 2 3 4 5 6

0 1 2 3 4 5 6

0

0

1 2 3 4 5 6

02

01

0

minus01

minus02

02

01

0

0

minus01

minus02

015

01

005

minus005

minus01

008

006

004

002

0

minus002

minus004

minus006

minus008

008

006

004

002

0

minus002

minus004

minus006

minus008

008

006

004

002

minus002

minus004

minus006

minus008

2

15

1

05

0

minus05

minus1

minus15

minus2

01

005

0

minus005

minus01Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at735

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

795

Hz (

mm

)

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at855

Hz (

mm

)

tran

sient

to st

eady

at755

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mTh

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

815

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

875

Hz (

mm

)

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at775

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

835

Hz (

mm

)

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at895

Hz (

mm

)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Figure 11 Theoretical tip displacement responses with the second-type mode shape of the beam from transient to steady state

the approximate position of 119909 = 1204mm This is justan approximate value because Eddy-Current Sensor cannotgive exact measurement results of the displacement at thejunction between the cantilever and the tip-mass This maycause that the amplitude level from theoretical discussionwashigher than from experimental measurement In additionthe amplitude of the instantaneous base displacement fromthe exciter may be not the fixed value 001mm at the startinginstant Therefore the maximum displacement responsesduring the transient state in the experimental measurementsmay shift away from the real experimental results

The theoretical responses with two improved types ofmode shapes and the experimental displacement responsesversus varying positions of beam and the frequencies atsteady state are shown in Figures 14ndash16 respectively Figures14 and 15 show two types of theoretical displacements atthe positions of 119909 = 364mm 574mm 784mm 994mmand 1204mmwith continuous varying frequencies In exper-iments the displacement responses were measured at thesame positions with theoretical calculations but with somecertain frequencies of 782Hz 792Hz 802Hz 807Hz81Hz 812Hz 817Hz 82Hz 83Hz and 84Hz as plottedin Figure 16 It is noticed that when the frequency of baseexcitation approached to the first natural frequency of thecantilever system the amplitude of forced vibration willrapidly increase and become maximum for the case when

the frequency of the excitation coincides with the naturalfrequency Table 3 lists the values of the first natural frequen-cies and the displacement responses from theoretical andexperimental results at some certain positions long the beamunder the first natural frequencies It can be seen that thetheoretical and experimental results are matched very close

Table 3 The first natural frequencies and the resonantdisplacements at some certain positions

In a word the theoretical results with two different modeshapes derived by two improved analytical methods haveextremely the same behaviors at transient and steady stateswith respect to varying frequencies from base excitationThedisplacement responses indicate close agreement betweenthe experimental and analytical results All of these resultsgive full verification for the improvement of the analyticalmethods in this paper utilized in recalculating the modeshapes of a cantilever beam when an offset distance existsbetween the gravity center of the tip-mass and the point ofattachment

5 Conclusions

This paper presents two improved analytical methods ofcalculation for natural frequencies and mode shapes of auniform cantilever beam carrying a tip-mass under an input

Shock and Vibration 13

005

0

minus005Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at735

Hz (

mm

)

0 1 2 3 4 5 6

Time (s)

0 1 2 3 4 5 6

0 1 2 3 4 5 6

02

01

0

minus01

minus02

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at795

Hz (

mm

)

0

015

01

005

minus005

minus01

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at855

Hz (

mm

)

0 1 2 3 4 5 6

Time (s)

008

006

004

002

0

minus002

minus004

minus006

minus008tran

sient

to st

eady

at755

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

m

2

15

1

05

0

minus05

minus1

minus15

minus2Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at815

Hz (

mm

)

0 1 2 3 4 5 6

008

006

004

002

0

minus002

minus004

minus006

minus008Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at875

Hz (

mm

)

0 1 2 3 4 5 6

01

005

0

minus005

minus01Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at775

Hz (

mm

)

0 1 2 3 4 5 6

Time (s)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

0 1 2 3 4 5 6

0 1 2 3 4 5 6

0

008

006

004

002

minus002

minus004

minus006

minus008Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at895

Hz (

mm

)

02

01

0

minus01

minus02

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at835

Hz (

mm

)Figure 12 Theoretical tip displacement responses with third-type mode shape of the beam from transient to steady state

Table 3 The first natural frequencies and the resonant displacements at some certain positions

Resonant frequency or positions The values of frequencies and displacement responses (Hz or mm) ErrorsTheory based on 119880(2)

1(119909) Theory based on 119880(3)

1(119909) Experiment

The 1st natural frequency 815 815 812 037119909 = 364mm 01285 01285 01277 062119909 = 574mm 03022 03022 03001 069119909 = 784mm 05279 05279 05246 063119909 = 994mm 07905 07905 07855 063119909 = 1204mm 10742 10742 10671 066

base excitation using forced vibration theory and the methodof separation of variables A simplified cantilever model byreplacing the tip-mass with an equivalent inertial force andinertial moment acting at the free end of the cantilever beamwas detailed In addition using the Dirac Delta Functionthe base transverse excitation the concentrated equivalentinertia force and inertia moment were represented in theform of distributed loads which can make the method ofseparation of variables be utilized properly In this case sometypical results of natural frequencies and mode shapes of thecantilever beam with a tip-mass were derived using the rein-forced method of separation of variables and the developedforced vibration respectively Furthermore based on modalanalysis method the transient and steady displacement

responses of the cantilever system with a tip-mass and a sinebase excitation were presented in which two different types ofmode shapes calculated by two improved analytical methodswere used Finally an experimental cantilever system wasfabricated and some practical displacement responses weremeasured

From the results of the typical natural frequencies andmode shapes we can easily found that the mode shapesfrom the first improved type were identical to the modeshapes from the second type whether the mode shapes fromthe lower order or from the higher order even the offsetdistance was large The comparing results of the transientand steady displacement responses from the theoreticalanalysis and experimental measurements further proved the

14 Shock and Vibration

005

0

minus0050 2 4 6 8 10 12

0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12

0 2 4 6 8 10 120 2 4 6 8 10 12

0 2 4 6 8 10 12 0 2 4 6 8 10 120 2 4 6 8 10 12

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at732

Hz (

mm

)

0

008

006

004

002

minus002

minus004

minus006

minus008

0

008

006

004

002

minus002

minus004

minus006

minus008

0

008

006

004

002

minus002

minus004

minus006

minus008

0

015

01

005

minus005

minus01

01

005

0

minus005

minus01

2

15

1

05

0

minus05

minus1

minus15

minus2

03

02

01

0

minus01

minus02

03

02

01

0

minus01

minus02

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at752

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at772

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at792

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at812

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at832

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at852

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at872

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at892

Hz (

mm

)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Figure 13 Experimental displacement responses at the position of 119909 = 1204mm from transient to steady state

12

1

08

06

04

02

0120 100 80 60 40 20

78

80

82

84Theo

retic

al st

eady

disp

lace

men

t (m

m)

Position at the beam (mm)Freq

uency

(Hz)

Figure 14 Theoretical steady-displacement responses versus posi-tions and frequencies with mode 119880(2)

1(119909)

improvements of the analytical methods for calculating themode shapes of the cantilever beam with a tip-mass anda base excitation In addition the amplitudes of the modeshapes decreased with the increasing mass value of the tip-mass when the moment of inertia and offset were constantsThe amplitude of the first-order mode shape increased but

12

1

08

06

04

02

0120 100 80 60 40 20

78

80

82

84Theo

retic

al st

eady

disp

lace

men

t (m

m)

Position at the beam (mm)Freq

uency

(Hz)

Figure 15 Theoretical steady-displacement responses versus posi-tions and frequencies with mode 119880(3)

1(119909)

the other nine mode shapes decreased with the increasingmoment of inertia when the mass and offset were set asconstants Conversely the amplitude of the first mode shapedecreased but the other ninemode shapes increased with theincreasing offset distance under the condition that the massand moment of inertia were constants Globally the effect

Shock and Vibration 15

12

1

08

06

04

02

0

Expe

rimen

tal s

tead

y di

spla

cem

ent (

mm

)

120 100 80 60 40 20Position at the beam (mm)

78

80

82

84Freq

uency

(Hz)

Figure 16 Experimental steady-displacement responses versuspositions and frequencies

from varying combinations of the mass and moment inertiaof the tip-mass and the offset distance for the lower-ordermode shapes was stronger than for the higher ones

Furthermore it is clearly noticed that the mass of tip-mass can reduce the values of the natural frequencies andthe increasing mass value gave the decreasing frequencieswhen the moment of inertia of the tip-mass and the offsetdistance were set as constants Similarly the moment ofinertia can further decrease the natural frequencies and thehigher moment of inertia resulted in the lower frequencieswhen the mass and the offset were constants Differently theoffset distance can reduce the first-order natural frequenciesbut increase the other nine order frequencies and the higheroffset distance led to the lower first-order natural frequenciesbut produced the higher other nine order frequencies whenthe mass and the moment of inertia were set to constant Inglobal view the influence from the varying mass momentof inertia and offset distance for the lower-order naturalfrequencies was larger than for the higher-order ones weatherthis influence was negative or positive

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the Natural Science Foundationsof China (no 51275380) and the Scientific and TechnologicalProject of Shaanxi Province (no 2012K06-36) The authorswould like to thank the reviewers for their valuable commentson this paper

References

[1] P A A Laura J L Pombo and E A Susemihl ldquoA note on thevibrations of a clamped-free beam with a mass at the free endrdquoJournal of Sound and Vibration vol 37 no 2 pp 161ndash168 1974

[2] B Rama Bhat andHWagner ldquoNatural frequencies of a uniformcantilever with a tip mass slender in the axial directionrdquo Journalof Sound and Vibration vol 45 no 2 pp 304ndash307 1976

[3] C W S To ldquoVibration of a cantilever beam with a baseexcitation and tip massrdquo Journal of Sound and Vibration vol83 no 4 pp 445ndash460 1982

[4] P A A Laura ldquoComments on lsquoVibration of a cantilever beamwith a base excitation and tip massrsquordquo Journal of Sound andVibration vol 88 no 4 p 569 1983

[5] C W S To ldquoDiscussion on lsquovibration of a cantilever beam witha base excitation and tip massrsquordquo Journal of Sound and Vibrationvol 97 no 2 pp 349ndash351 1984

[6] R G Jacquot ldquoFurther comments on lsquoVibration of a cantileverbeam with a base excitation and tip massrsquordquo Journal of Sound andVibration vol 93 no 2 pp 312ndash313 1984

[7] S AMousavi Lajimi andG R Heppler ldquoComments on lsquonaturalfrequencies of a uniform cantilever with a tip mass slender inthe axial directionrsquordquo Journal of Sound and Vibration vol 331 no12 pp 2964ndash2968 2012

[8] R B Bhat ldquoAuthors response to the comments on lsquonaturalfrequencies of a uniform cantilever with a tip mass slender inthe axial directionrsquordquo Journal of Sound and Vibration vol 331 no12 pp 2962ndash2963 2012

[9] E Esmailzadeh and G Nakhaie-Jazar ldquoPeriodic behavior ofa cantilever beam with end mass subjected to harmonic baseexcitationrdquo International Journal of Non-Linear Mechanics vol33 no 4 pp 567ndash577 1998

[10] H Abramovich and O Hamburger ldquoVibration of a cantilevertimoshenko beam with a tip massrdquo Journal of Sound andVibration vol 148 no 1 pp 162ndash170 1991

[11] B Pratiher ldquoVibration control of a transversely excited can-tilever beam with tip massrdquo Archive of Applied Mechanics vol82 no 1 pp 31ndash42 2012

[12] M Kim M Hoegen J Dugundji and B L Wardle ldquoModelingand experimental verification of proof mass effects on vibrationenergy harvester performancerdquo Smart Materials and Structuresvol 19 Article ID 045023 2010

[13] Y T Li and R Wong ldquoIntegral and series representations ofthe Dirac delta functionrdquo Communications on Pure and AppliedAnalysis vol 7 no 2 pp 229ndash247 2008

[14] M Al-Gwaiz Sturm-LiouvilleTheory and its Applications chap-ter 3 Springer London UK 2008

[15] F Bao-lian ldquoGeneralized reciprocal theorems and their appli-cationsrdquo Applied Mathematics and Mechanics vol 23 no 2 pp203ndash210 2001

[16] C Cai H Zheng M Khan and K HungModeling of MaterialDamping Properties in ANSYS Defense Systems Division Insti-tute of High Performance Computing 2002

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Shock and Vibration

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International Journal of

Shock and Vibration 3

ri

rcCg

mi

aiac

F(1)gi

F(1)g

o998400

M(1)g

(a)

ri

rcCg

mi

acF(2)gi

F(2)g

o998400

a120591i

M(2)g

120585

(b)

Figure 2 Simplification of the tip-mass to point 1199001015840 under (a) translational motion and (b) rotational motion

y

x

b(t) ua(x t)

FgMg

EI m L

Figure 3 The simplified model of cantilever with tip-mass beingequated by inertial force and moment

force 119865119892and inertia moment 119872

119892can be represented in the

form of distributed loads as

119891119886(119909 119905)

= 119865119892120575 (119909 minus 119871)

= [119872

1205972119906119886(119909 119905)

1205971199052

+ 119903119888119872

1205972

1205971199052(

120597119906119886(119909 119905)

120597119909

)] 120575 (119909 minus 119871)

119872119886(119909 119905)

= 119872119892120575 (119909 minus 119871)

= [119903119888119872

1205972119906119886(119909 119905)

1205971199052

+ (119869 + 1199032

119888119872)

1205972

1205971199052(

120597119906119886(119909 119905)

120597119909

)]

times

119889120575 (119909 minus 119871)

119889119909

(4)

Therefore employing variational method the dynamicmotion equation of the simplified model of the cantileverbeam with a tip-mass under base transverse excitation isobtained as

119864119868

1205974119906119886(119909 119905)

120597119909

+ 119898

1205972119906119886(119909 119905)

120597119905

= 119891119886(119909 119905) minus 119872

119886(119909 119905) (5)

where 119864119868 and119898 denote the flexural rigidity and the mass perunit length of the beam respectively

The absolute displacement can be considered as thesum of the relative displacement and the base displacementexcitation

119906119886(119909 119905) = 119906rel (119909 119905) + 119887 (119905) (6)

Substituting (6) into (5) gives the new form of the dynamicmotion equation as

119864119868

1205974119906rel (119909 119905)

120597119909

+ 119898

1205972119906rel (119909 119905)

120597119905

= [119872

1205972119906rel (119909 119905)

1205971199052

+ 119903119888119872

1205972

1205971199052(

120597119906rel (119909 119905)

120597119909

)] 120575 (119909 minus 119871)

+ [119903119888119872

1205972119906rel (119909 119905)

1205971199052

+ (119869 + 1199032

119888119872)

1205972

1205971199052(

120597119906rel (119909 119905)

120597119909

)]

times

119889120575 (119909 minus 119871)

119889119909

+ [119872120575 (119909 minus 119871) + 119903119888119872

119889120575 (119909 minus 119871)

119889119909

minus 119898]

1198892119887 (119905)

1198891199052

(7)

In this case the simplified model of the cantilever beamunder base excitation can be considered as a new analyticalmodel with the dynamic differential equation of (8) and theboundary conditions of (9) through (12)

119864119868

1205974119906119886(119909 119905)

120597119909

+ 119898

1205972119906119886(119909 119905)

120597119905

= [119872120575 (119909 minus 119871) + 119903119888119872

119889120575 (119909 minus 119871)

119889119909

minus 119898]

1198892119887 (119905)

1198891199052

(8)

at 119909 = 0

119906119886(119909 119905) = 0 (9)

119889

119889119909

119906119886(119909 119905) = 0 (10)

4 Shock and Vibration

at 119909 = 119871

119864119868

1198893

1198891199093119906119886(119909 119905) = [119872

1205972119906119886(119909 119905)

1205971199052

+ 119903119888119872

1205972

1205971199052(

120597119906119886(119909 119905)

120597119909

)]

(11)

119864119868

1198892

1198891199092119906119886(119909 119905)

= minus [119903119888119872

1205972119906119886(119909 119905)

1205971199052

+ (119869 + 1199032

119888119872)

1205972

1205971199052(

120597119906119886(119909 119905)

120597119909

)]

(12)

The solutions of (8) subjected to four boundary condi-tions and two initial conditions can be obtained convenientlyby the traditional method of separation of variables Themethod regards the response as a superposition of the systemeigen-functionsmultiplied by corresponding time dependentgeneralized coordinates Of course this necessitates firstobtaining the solution of the system eigen-value problemTurning attentions to the corresponding eigen-value problemand considering the free vibration characterized by setting119887(119905) = 0 the solution of (8) becomes separable in space andtime Letting

119906119886(119909 119905) = 119880 (119909) 119879 (119905) (13)

and substituting (13) into the free vibration form of (8) twodependent equations are obtained using the separation ofvariables method as

119889

1198891199052119879 (119905) + 120596

2119879 (119905) = 0 (14)

1198894

1198891199094119880 (119909) minus 120573

4119880 (119909) = 0 (15)

where 1205962 indicates the angular frequency and 1205734 =

1198981205962(119864119868) In this case the four boundary conditions will be

changed as

at 119909 = 0

119880 (119909) = 0 (16)

119889

119889119909

119880 (119909) = 0 (17)

at 119909 = 119871

119864119868

1198893

1198891199093119880 (119909) = minus [120596

2119872119880(119909) + 119903

1198881198721205962(

119889

119889119909

119880 (119909))] (18)

119864119868

1198892

1198891199092119880 (119909) = 119903

1198881198721205962119880 (119909) + (119869 + 119903

2

119888119872)1205962(

119889

119889119909

119880 (119909))

(19)

The general solution of (15) can be easily derived to be

119880 (119909) = 1198621cos (120573119909) + 119862

2sin (120573119909) + 119862

3cosh (120573119909)

+ 1198624sinh (120573119909)

(20)

where 119862119899(119899 = 1 2 3 4) are constants of integration In

order to evaluate three of these constants in terms of thefourth and derive the characteristic equation the boundaryconditions (16) to (19) must be used Indeed substituting (20)into (16) through (19) gives the characteristic equation andthe mode shapes as

1 + 119888119888ℎ minus 120601119906 (119904119888ℎ minus 119888119904ℎ)

minus 1198771[119904119888ℎ + 119888119904ℎ minus 120601119906 (1 minus 119888119888ℎ)]

minus 21198772119904119904ℎ minus 119877

2

2(1 minus 119888119888ℎ) = 0

(21)

119880 (119909) = 1198622sin (120573119909) minus 119904ℎ (120573119909) minus 120582 [cos (120573119909) minus 119888ℎ (120573119909)]

(22)

where 120582 = 1198601211986011and the constant 119862

2is an arbitrary value

and in which

11986011= minus(1 + 119877

2)119888 + (119877

2minus 1)119888ℎ + 119877

1(119904 + 119904ℎ) 119860

12=

1198771(119888ℎ minus 119888) minus (1 + 119877

2)119904 + (119877

2minus 1)119904ℎ

119906 = 120573119871 119904 = sin(119906) 119888 = cos(119906) 119904ℎ = sinh(119906) 119888ℎ =cosh(119906)

120601 = 119872(119898119871) 1198771= 120601119906

3[(119870119871)

2+ (119903119888119871)2] 1198772=

1206011199062(119903119888119871) 119870 = radic119869119872

23TheDevelopedMethod of Forced Vibration If we treat thebase displacement excitation as a kind of boundary conditionthen the differential equation of motion and four boundaryconditions of the cantilever system can be written as

119864119868

1205974119906119886(119909 119905)

120597119909

+ 119898

1205972119906119886(119909 119905)

120597119905

= 0 (23)

at 119909 = 0

119906119886(0 119905) = 119887 (119905) (24)

119889

119889119909

119906119886(0 119905) = 0 (25)

at 119909 = 119871

119864119868

1198893

1198891199093119906119886(119909 119905)

= [119872

1205972119906119886(119909 119905)

1205971199052

+ 119903119888119872

1205972

1205971199052(

120597119906119886(119909 119905)

120597119909

)]

(26)

119864119868

1198892

1198891199092119906119886(119909 119905)

= minus [119903119888119872

1205972119906119886(119909 119905)

1205971199052

+ (119869 + 1199032

119888119872)

1205972

1205971199052(

120597119906119886(119909 119905)

120597119909

)]

(27)

Shock and Vibration 5

Under the simple harmonic excitation 119890119895120596119905 the steady statesolution of (23) can be regarded with the form as

119906119886(119909 119905) = 119880

119886(119909 120596) 119890

119895120596119905minus120593

= [1198611cos (120573119909) + 119861

2sin (120573119909)] 119890119895120596119905minus120593

+1198613cosh (120573119909) + 119861

4sinh (120573119909)]

(28)

where 119861119899(119899 = 1 2 3 4) are arbitrary constants the

parameter 120573 is defined by 1205734 = 1198981205962(119864119868) 119895 is the imaginaryunit and 120593 is the lagging phase

We recalculated the natural frequencies and the modeshapes directly by substituting (28) into four boundaryconditions (24) through (27) and obtained the characteristicequation and displacement response which are different from[3]

119863 = 1 + 119888119888ℎ minus 120601119906 (119904119888ℎ minus 119888119904ℎ)

minus 1198771[119904119888ℎ + 119888119904ℎ minus 120601119906 (1 minus 119888119888ℎ)]

minus 21198772119904119904ℎ minus 119877

2

2(1 minus 119888119888ℎ)

119880 (119909) = (

119861

2119863

) 1198651015840

1119906cos (120573119909) + 1198651015840

2119906cosh (120573119909)

+1198651015840

3119906[sin (120573119909) minus sinh (120573119909)]

(29)

where 119861 is defined by 119887(119905) = 119861119890119895120596119905 and in which

1198651015840

1119906= 1+119888119888ℎminus119904119904ℎminus2120593119906119904119888ℎminus2119877

1119888119904ℎ+(119877

1120593119906minus119877

2

2)(1minus

119888119888ℎ + 119904119904ℎ) minus 21198772(119888119888ℎ + 119904119904ℎ) =119865

1119906minus 21198772(119888119888ℎ + 119904119904ℎ)

1198651015840

2119906= 1+119888119888ℎ+119904119904ℎ+2120593119906119888119904ℎminus2119877

1119904119888ℎ+(119877

1120593119906minus119877

2

2)(1minus

119888119888ℎ minus 119904119904ℎ) + 21198772(119888119888ℎ minus 119904119904ℎ) =119865

2119906+ 21198772(119888119888ℎ minus 119904119904ℎ)

1198651015840

3119906= (119904119888ℎ + 119888119904ℎ)(1 minus 119877

1120593119906 +119877

2

2) + 2[120593119906119888119888ℎ minus 119877

1119904119904ℎ minus

1198772(119904119888ℎ minus 119888119904ℎ)] = 119865

3119906

It is obvious that the characteristic equation derived by thereinforced method of separation of variables is equal tothe equation obtained by the developed method of forcedvibration

Themode shapes cannot be obtained directly because theboundary conditions contain the parameter1205962 However thisproblem can be successfully solved by Dirac Delta Function[13] Thus the mode shapes may be obtained by the Sturm-Liouville theory [14] and expressed as

119880119894(119909) = (

1

2

) 1198651015840

1119906119894119865 cos (120573

119894119909) + 119865

1015840

2119906119894cosh (120573

119894119909)

+ 1198651015840

3119906119894[sin (120573

119894119909) minus sinh (120573

119894119909)]

(30)

3 Computed Results and Comparisons

In this section three types of the natural mode shapes ofthe cantilever beam with a tip-mass were computed and

compared The first one is called the undeveloped modeshapes which is directly obtained from [3] by the expression

119880(1)

119894(119909) = (

1

2

) 1198651119894119888

cos(119906119894119909

119871

) + 1198652119894119888

cosh (119906119894119909

119871

)

+ 1198653119894119888[sin(

119906119894119909

119871

) minus sinh(119906119894119909

119871

)]

(31)

where 1198651119894119888

= 1198651119894119888ℎ(119906119894) 1198652119894119888

= 1198652119894119888ℎ(119906119894) and 119865

3119894119888=

1198653119894119888ℎ(119906119894) in which 119865

1119894 1198652119894 and 119865

3119894are the functions 119865

1119906 1198652119906

and 1198653119906

with the variable 119906 replaced by 119906119894 respectively

The second one is based on the expression of thedeveloped mode shapes derived in this paper Let (30) bemultiplied by 21198651015840

3119894to give the second mode shapes as

119880(2)

119894(119909) = sin(

119906119894119909

119871

) minus sinh(119906119894119909

119871

) + (

1198651015840

1119894

1198651015840

3119894

) cos(119906119894119909

119871

)

+ (

1198651015840

2119894

1198651015840

3119894

) cosh (119906119894119909

119871

)

(32)

At last the arbitrary constant 1198622in (22) was chosen as

constant 1 to give the third type of the natural mode shapes as

119880(3)

119894(119909) = sin(

119906119894119909

119871

) minus 119904ℎ (

119906119894119909

119871

)

minus 120582 [cos(119906119894119909

119871

) minus 119888ℎ (

119906119894119909

119871

)]

(33)

Before the mode shapes are determined the roots ofthe characteristic equation must be calculated firstly Thecharacteristic equation (21) or 119863 = 0 can be solved by twosteps First approximate roots were pointed out by applyinggraphing method Then the approximate roots were iteratedby the bisection method to give the exact characteristic rootswith given error All these processes were finished usingMATLAB software

Because the condition that the gravity center of the tip-mass coincided with the point of the end attachment hasbeen verified in [6] we focus on the condition when theoffset distance is not zero Table 1 shows the first five roots ofthe characteristic equations from five samples with differentvalues of 120601 119896119871 and 119889119871 It can be clearly noticed thatthe larger value of 119889119871 resulted in the less value of the firstcharacteristic root but the larger values of the other fourcharacteristic roots when 120601 and 119896119871 were set as constants

Once the roots of the characteristic equation have beenobtained three types of the first five mode shapes of thecantilever structure from the five given samples can becalculated Figures 4 5 6 7 and 8 show the first five modeshapes of the cantilever with various combinations of 120601 119896119871and 119889119871

6 Shock and Vibration

Table 1 Typical roots of characteristic equation with variables 120601 119896119871 and 119889119871

Sample 120601 119870119871 119889119871

Roots of characteristic equation1199061

1199062

1199063

1199064

1199065

1 4 4 0 035211 129234 477818 788401 11017802 4 4 05 034814 130637 477952 788472 11018273 4 4 1 034200 132825 478225 788634 11019404 4 4 2 032544 138983 479177 789233 11023685 4 4 4 028806 154543 482543 791455 1103990

003002001

0

0 05 1 0 05 1 0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

000000

050000

100000

i = 1

000000

050000

100000

i = 1

000000 000000

000000

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

000000

050000

100000

i = 2

000000

050000

100000

i = 2

minus186571 times 10minus9

000792

003152

minus198560 times 10minus8

minus9418658

minus18545378

0

minus100

minus2000 05 1

0 05 1

0 05 1

0 05 1

1050

times104 i = 3

000000

050000

100000

i = 3

000000

050000

100000

i = 3

786191 times 10minus6

111026 times 104

minus686262 times 103

50

minus5

times105 i = 4000000

050000

100000

i = 4

000000

050000

100000

i = 4

i = 5000000

050000

100000

i = 5

000000 000000

050000

100000

i = 5times106

20

minus2

000017

minus271300 times 106

minus855944 times 104

1

05

0

1

05

0

03

02

01

0

03

02

01

0

151

050

151

050

2

0

minus2

2

0

minus2

210

minus1

210

minus1

minus493414 times 10minus8

032943

131050

367824 times 10minus11

017448

034354

113682 times 10minus10

160541

minus009923

457130 times 10minus11

minus002114

006144

866744 times 10minus11

minus140614

minus004436

032943

131050

017448

034354

160541

minus009923

000000

minus002114

006144

minus140614

minus004436

Notes at xL Notes at xL Notes at xLValues ValuesValues

Ui(x)

xL

Ui(x)

xL

Ui(x)

xL

minus233509 times 10minus5

107963 times 104

minus313859 times 104

First-type mode shapes U(1)i

(x) Second-type mode shapes U(2)i

(x) Third-type mode shapes U(3)i

(x)

Figure 4 Three types of the first five mode shapes with the combinations of 120601 = 4 119896119871 = 4 and 119889119871 = 0

It is obvious that when 119889119871 = 0 the first type of naturemode shapes 119880(1)

119894(119909) derived by the undeveloped method

turned far away from the third-type mode shapes 119880(3)119894(119909)

derived by the reinforced method of separation of variablesIn addition the larger value of 119889119871 resulted in the moreinconsistent mode shapes However the second type of modeshapes 119880(2)

119894(119909) obtained by the developed method of forced

vibration seemed in good agreement with the third onesFurthermore the amplitude of the first mode shape

decreased but the other four mode shapes increased with theincreasing value of the 119889119871 when 119896119871 and 120601 were fixed to be

constants The effect from the varying value of 119889119871 for thelower mode shapes was always stronger than for the higher

4 Experimental Verification

In order to experimentally explain and compare two typesof natural mode shapes derived by two improved methodsabove the transient and steady responses of the cantileversystem under an input base excitation should be calculatedfirst This whole calculation can be finished by method ofmodal analysis But above all the orthogonality property of

Shock and Vibration 7

0 05 1

0 05 1

Ui(x)

xL

Ui(x)

xL

Ui(x)

xL

014013012011

0

minus100

minus200

000000

050000

100000

i = 1

000000

050000

100000

i = 2

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

011044

011840

014143

533123

minus9287788

minus18666064

0 05 1

1050

times104

000000

050000

100000

i = 3minus8449356

107519 times 105

minus970528 times 103

0 05 1

50

minus5

times105

000000

050000

100000

i = 422493344

139452 times 104

139108 times 105

0 05 1

40

minus4minus8

times106

000000

050000

100000

i = 5minus42806116

minus266045 times 106

minus765307 times 106

0 05 1

000000

050000

100000

i = 106

04

02

0

392120 times 10minus9

017490

069258

0 05 1

06

04

02

0

0 05 1

000000

050000

100000

i = 2547973 times 10minus11

017881

034942

03

02

01

00 05 1

03

02

01

0

0 05 1

000000

050000

100000

i = 3minus505260 times 10minus11

160517

minus010133

151

050

0 05 1

151

050

0 05 1

000000

050000

100000

i = 4minus776048 times 10minus11

minus002162

006261

2

0

minus20 05 1

2

0

minus2

0 05 1

000000

050000

100000

i = 5663249 times 10minus11

minus140614

minus004516

210

minus1

000000

000000

000000

000000

017490

069258

017881

034942

160517

minus010133

minus002162

006261

000000

050000

100000

i = 5

0 05 1

210

minus1

000000

minus140614

minus004516

Notes at xL Notes at xL Notes at xLValues ValuesValuesFirst-type mode shapes U(1)i

(x) Second-type mode shapes U(2)i

(x) Third-type mode shapes U(3)i

(x)

Figure 5 Three types of the first five mode shapes with the combinations of 120601 = 4 119896119871 = 4 and 119889119871 = 05

034

033

032

031

0 05 1 0 05 1 0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1 0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0

minus100

minus200

10

5

0

times104

50

minus5

times105

times106

0minus5minus10minus15

Ui(x)

xL

Ui(x)

xL

Ui(x)

xL

04

02

0

04

02

0

04

02

0

04

02

0

151

050

151

050

2

0

minus2

2

0

minus2

210

minus1

210

minus1

000000

050000

100000

i = 1

000000 000000

050000

100000

i = 1

030899

031712

033934

000000

050000

100000

i = 2861965

minus9105604

minus18440224

000000

050000

100000

i = 3minus10637851

973291 times 104

minus123754 times 104

000000

050000

100000

i = 419388076

168672 times 104

311790 times 105

000000

050000

100000

i = 5minus19120922

minus244086 times 106

minus152554 times 107

000000

050000

100000

i = 1

331605 times 10minus9

012125

047836

000000

050000

100000

i = 2

minus550109 times 10minus11

018658

036219

000000

050000

100000

i = 3

minus175471 times 10minus11

160459

minus010631

000000

050000

100000

i = 4

109777 times 10minus10

minus002275

006562

000000

050000

100000

i = 5

836395 times 10minus11

minus140614

minus004731

012125

047836

000000 000000

050000

100000

i = 2

018658

036219

000000 000000

050000

100000

i = 4

minus002275

006562

000000 000000

050000

100000

i = 5

minus140614

minus004731

000000 000000

050000

100000

i = 3

160459

minus010631

Notes at xL Notes at xL Notes at xLValues ValuesValuesFirst-type mode shapes U(1)i

(x) Second-type mode shapes U(2)i

(x) Third-type mode shapes U(3)i

(x)

Figure 6 Three types of the first five mode shapes with the combinations of 120601 = 4 119896119871 = 4 and 119889119871 = 1

8 Shock and Vibration

088087086085

0 05 1 0 05 1 0 05 1

0 05 1 0 05 1

0 05 10 05 1

0 05 1 0 05 1

0 05 1 0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0

minus50

minus100

minus150

6420

minus2

times104

840

minus4

times105

0minus1minus2minus3

times107

0302010

03

02

01

0

04

02

0

04

02

0

151

050

151

050

210

minus1

210

minus1

210

minus1

210

minus1

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

084906

085736

087731

minus1308377

minus9308045

minus16072843

63490847

565642 times 104

minus172997 times 104

minus329762092

230021 times 104

667852 times 105

946242416

minus149912 times 106

minus306922 times 107

minus995048 times 10minus10

007811

030669

511030 times 10minus11

021118

040600

minus114494 times 10minus10

160244

minus012466

688252 times 10minus11

minus002686

007708

183045 times 10minus11

minus140613

minus005561

000000

000000

000000

000000

000000

007811

030669

021118

040600

160244

minus012466

minus002686

007708

minus140613

minus005561

Ui(x)

xL

Ui(x)

xL

Ui(x)

xL

Notes at xL Notes at xL Notes at xLValues ValuesValuesFirst-type mode shapes U(1)i

(x) Second-type mode shapes U(2)i

(x) Third-type mode shapes U(3)i

(x)

Figure 7 Three types of the first five mode shapes with the combinations of 120601 = 4 119896119871 = 4 and 119889119871 = 2

the natural mode shapes should be derived In fact usingBetti theorem [15] the orthogonality properties with respectto mass and stiffness can be obtained respectively as follows

int

119871

0

119898119880(119896)

119903(119909)119880(119896)

119894(119909) 119889119909 +119872119880

(119896)

119903(119871)119880(119896)

119894(119871)

+ 119903119888119872119880(119896)

119894(119871)

119889119880(119896)

119903(119871)

119889119909

+ 119903119888119872119880(119896)

119903(119871)

119889119880(119896)

119894(119871)

119889119909

+ (119869 + 1199032

119888119872)

119889119880(119896)

119894(119871)

119889119909

119889119880(119896)

119903(119871)

119889119909

= (120575(119896)

119903119894)

2

(119896 = 1 2 119903 119894 = 1 2 )

(34)

int

119871

0

119864119868

1198892119880(119896)

119894(119909)

1198891199092

1198892119880(119896)

119903(119909)

1198891199092

119889119909 = (120596119903120575(119896)

119903119894)

2

(119896 = 1 2 119903 119894 = 1 2 )

(35)

In this case the normalized mode shapes can be defined as

119880(119896)

119903119873(119909) =

119880(119896)

119903(119909)

120575(119896)

119903119903

(36)

where

(120575(119896)

119903119903)

2

= int

119871

0

119898[119880(119896)

119903(119909)]

2

119889119909 +119872[119880(119896)

119903(119871)]

2

+ 2119903119888119872119880(119896)

119903(119871) [

119889

119889119909

119880(119896)

119903(119871)]

+ (119869 +1198721199032

119888) [

119889

119889119909

119880(119896)

119903(119871)]

2

(37)

41 Theoretical Responses of the Cantilever System with a BaseSine Excitation According to the method of modal analysisthe solution of (8) has the form of

119906119886(119909 119905) =

infin

sum

119903=1

119880(119896)

119903119873(119909) 119902119903(119905) (38)

So inserting (38) into (8) gives

119864119868

infin

sum

119903=1

1198894119880(119896)

119903119873(119909)

1198891199094

119902119903(119905) + 119898

infin

sum

119903=1

119880(119896)

119903119873(119909)

1198892119902119903(119905)

1198891199052

= [119872120575 (119909 minus 119871) + 119903119888119872

119889120575 (119909 minus 119871)

119889119909

minus 119898]

1198892119887 (119905)

1198891199052

(39)

Considering the orthogonality properties of equations (34)and (35) let (39) be multiplied by119880(119896)

119894119873(119909) and integrated over

Shock and Vibration 9

192

191

19

0 05 1 0 05 1

0 05 1 0 05 1

0 05 1

0 05 10 05 1

0 05 10 05 1

0 05 1 0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0

minus200

minus400

minus600

0minus5minus10

times104

1

0

minus1

times106

times107

0minus2minus4minus6

02

01

0

02

01

0

06

04

02

0

06

04

02

0

15

1

05

0

15

1

05

0

210

minus1

210

minus1

210

minus1

210

minus1

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

000000

050000

100000

i = 1

000000

050000

100000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

189913

190621

192051

minus56420429

minus28138349

4723398

134513 times 104

minus110859 times 105

minus260952 times 104

minus615688 times 104

496921 times 104

145091 times 106

168417 times 105

241806 times 106

minus633899 times 107

137503 times 10minus9

004964

019458

minus238241 times 10minus11

028530

054402

159443

minus019150

minus283947 times 10minus11

minus004212

012050

771494 times 10minus11

minus140605

minus008755

000000

000000

000000 000000

000000

000000

004964

019458

028530

054402

159443

minus019150

minus004212

012050

minus140605

minus008755

Ui(x)

xL

Ui(x)

xL

Ui(x)

xL

Notes at xL Notes at xL Notes at xLValues ValuesValues

minus387890 times 10minus11

First-type mode shapes U(1)i

(x) Second-type mode shapes U(2)i

(x) Third-type mode shapes U(3)i

(x)

Figure 8 Three types of the first five mode shapes with the combinations of 120601 = 4 119896119871 = 4 and 119889119871 = 4

the domain 0 lt 119909 lt 119871 to give a set of independent ordinarydifferential equations as

1198892119902119894(119905)

1198891199052

+ 1205962

119894119902119894(119905) = 119891

(119896)

119894(119905) (40)

Considering the properties of Dirac Delta Function

int

119871

0

119880(119896)

119894119873(119909) 120575 (119909 minus 119871) 119889119909 = 119880

(119896)

119894119873(119871)

int

119871

0

119880(119896)

119894119873(119909)

119889120575 (119909 minus 119871)

119889119909

119889119909 = minus

119889119880(119896)

119894119873(119871)

119889119909

(41)

the generalized forces associated with the generalized coordi-nates 119902

119894(119905) can be obtained as

119891(119896)

119894(119905)

= [119872119880(119896)

119894119873(119871) minus 119903

119888119872

119889119880(119896)

119894119873(119871)

119889119909

minus 119898int

119871

0

119880(119896)

119894119873(119909) 119889119909]

1198892119887 (119905)

1198891199052

(42)

Furthermore considering the mechanical damping asRayleigh damping and ignoring the part of mass dampingthe damping coefficient 120577 of the bimorph beam can be deter-mined by the modal damping ratio which can be obtained

by experiment [16] The independent ordinary differentialequations with damping then can be written as

1198892119902119894(119905)

1198891199052

+ 2120596119894120577

119889119902119894(119905)

119889119905

+ 1205962

119894119902119894(119905)

= [119872119880(119896)

119894119873(119871) minus 119903

119888119872

119889119880(119896)

119894119873(119871)

119889119909

minus 119898int

119871

0

119880(119896)

119894119873(119909) 119889119909]

1198892119887 (119905)

1198891199052

(43)

Now (43) can be solved using Laplace transform Applyinga sine excitation 119887(119905) = 119861 sin(120596119905) to the base of cantileverstructure under zero initial conditions the transfer functionof the independent differential equations for the cantileverbeam with tip-mass under base displacement excitation canbe written as

119866119894(119904)

=

119876119894(119904)

119861119894(119904)

=

(minus1205962)[119872119880

(119896)

119894119873(119871)minus119903

119888119872(119889119880

(119896)

119894119873(119871) 119889119909)minus119898int

119871

0119880(119896)

119894119873(119909) 119889119909]

1199042+ 2120596119894120577119904 + 120596

2

119894

(44)

10 Shock and Vibration

where 119876119894(119904) and 119861

119894(119904) are the Laplace transforms of 119902

119894(119905) and

119887(119905) respectively The inverse transformation of the transferfunction will be further obtained as

119892119894(119905)

=

(minus1205962)[119872119880

(119896)

119894119873(119871)minus119903

119888119872(119889119880

(119896)

119894119873(119871) 119889119909)minus119898int

119871

0119880(119896)

119894119873(119909) 119889119909]

120596119894radic1 minus 120577

2

times 119890minus120596119894120577119905 sin(120596

119894radic1 minus 120577

2119905)

(45)

In this case the solutions of (43) can be solved by convolutiontheorem as

119902119894(119905) = 119887 (119905) lowast 119892

119894(119905) = int

119905

0

119887 (120591) 119892119894(119905 minus 120591) 119889120591 (46)

Finally substituting the result of (46) into (38) and after somelengthy algebraic manipulation the absolute displacementresponse of the cantilever beam with tip-mass under basetransverse excitation can be obtained as

119906119886(119909 120596 119905) =

infin

sum

119894=1

119880(119896)

119894119873(119909)

119861 [119872119880(119896)

119894119873(119871) minus 119903

119888119872(119889119880

(119896)

119894119873(119871) 119889119909) minus 119898int

119871

0119880(119896)

119894119873(119909) 119889119909]

radic1205964+ 2 (2120577

2minus 1) 120596

21205962

119894+ 1205964

119894

times [

1205963

120596119894radic1 minus 120577

2

119890minus120596119894120577119905 sin(120596

119894radic1 minus 120577

2119905 + 1206011) + 1205962 sin (120596119905 + 120601

2)]

(47)

where 1206011and 120601

2are determined by tan(120601

1) = 2120577radic1 minus 120577

2

((120596120596119894)2+ 21205772minus 1) and tan(120601

2) = minus2120577(120596120596

119894)(1 minus (120596120596

119894)2)

It can be seen that the solution consists of two parts abouttimeThe first term represents free vibration which will decaywith time exponentially The second term depending on theexcitation represents the forced vibration of the system

42 Experimental Responses and Comparisons with the Theo-retical Results In order to explain the absolute displacementresponse of the cantilever structure under base excitationexperimentally a typical cantilever beam in which theelongated beam was inserted and fixed into the tip-mass tomake sure the gravity center of tip-mass was collinear withthe beam was fabricated (as shown in Figure 9) The chosengeometric and physical properties of the cantilever beamincluding the tip-mass are given in Table 2

The complete experimental devices are shown inFigure 10 The electromagnetic vibrating table type SW-TFAcontrolled by a programmable controller type SW-F-2TF-2was used to create a specific harmonic input excitationThe cantilever beam with a tip-mass was clamped onthe vibrating top platform Eddy-Current Sensors typeDM6008 and ZA210800 were used to measure the inputdisplacement from base vibrating exciter and the absolutedynamic displacement from the beam respectively Allmeasurements from cantilever beam and vibrating exciterwere connected to Digital Oscilloscope type DS1204Band the Industrial Computer including a PC-based DataAcquisitionModule from ADLINK company type DAQ2213The Digital Oscilloscope and Industrial Computer displayedand processed the measurement results using the DAQPilotsoftware and the results from subsequent analysis using theMATLAB software

Table 2 Characteristic properties of the cantilever beam includingtip-mass

Physical properties Beam Tip-massYoungrsquos modulus 119864 (GPa) 69 mdashDensity 120588

119904or 120588119898(kgm3) 2630 7753

Length 119871 or 119871119898(mm) 1204 156

Thickness 119905119904or 119905119898(mm) 49 295

Width 119908 (mm) 102 102Total mass119872 (g)a mdash 3383Offset 119903

119888(mm)b mdash 801

Rotary inertia 119869 (kgsdotm2)b c mdash 333 times 10minus6aThe total mass of the tip-mass including the elongated part of beam wascalculated by119872 = 119871

119898119905119898119908120588119898+ 119871119904119905119904119908(120588119904minus 120588119898) where 119871

119904= 10mm

bThe offset distance and the rotary inertia were calculated according to thegeometry and material properties of the beam and tip-mass using equations119903119888= (120588

1198981199051198981198712

119898minus 1205881198981199051199041198712

119904+ 1205881199041199051199041198712

119904)(2[120588

119898119905119898119871119898+ 119871119904119905119904(120588119904minus 120588119898)])

119869 = 120588119898119871119898119905119898119908 [(112)(119871

2

119898+ 1199052

119898) + ((12)119871

119898minus 119903119888)2] + 119871

119904119905119904119908(120588119904minus

120588119898)[(112)(119871

2

119904+ 1199052

119904) + (119903119888minus (12)119871

119904)2]

cThe corresponding parameters 120601 119896119871 and 119903119888119871 were calculated as follows

120601 = 119872(120588119904119905119904119908) = 21377 119896119871 = radic119869119872119871 = 00824 and 119903

119888119871 = 00665

The analytical model defined by (47) in which two typesof mode shapes derived by two improved analytical methodsare utilized can be used to analyze the steady and transientdisplacement responses of the cantilever beam with a tip-mass and an input base excitation based on the chosenvarying frequencies time and positions along the lengthof the beam Typically the displacement responses frompositions along 119909-axis at 119909

1= 364mm 119909

2= 574mm

1199093

= 784mm 1199094

= 994mm and 1199095

= 1204mmoff the origin point (as shown in Figure 9) were measured

Shock and Vibration 11

y

b(t)

x1

x2

x3

x4

x5

Ls

L

t s

t m x

Lm

rc

Cg

o998400

Figure 9 Typical cantilever beam with a tip-mass

1 2

3

56

7

8

4

Tip-massCantilever beam

(7) Cantilever with a tip-mass

(1) Industrial computer including data acquisitionmodule type DAQ2213

(2) Digital oscilloscope type DS1204B(3) Programmable controller type SW-F-FTF-2(4) Power supply type DP1308A(5) Driver modules for Eddy-current sensors(6) Electromagnetic vibrating table type SW-TFA

(8) Eddy-current sensors type DM6008 and ZA210800

Figure 10 Complete experimental setup

by Eddy-Current Sensors The amplitude of the input sine-displacement excitation for analysis was measured as 119861 =

001mm and the damping ratio for the transverse vibrationaround the fundamental resonant frequency was found to be120577 = 00025 by Logarithmic Decrement Method

In the following paragraphs we focus on the transientand steady responses of the cantilever system with a sine baseexcitation

Figures 11 and 12 show the theoretical displacementresponses at the free end of the beam from transient tosteady state with two improved mode shapes above at thefirst resonant frequency It can be seen obviously that thedisplacement responses with two improved mode shapeswe proposed are almost identical This richly proves thatthe first-type mode shapes reformulated by the reinforcedmethod of separation of variables are equal to the second-type mode shapes derived by the developed method offorced vibration In addition it can be further obtainedthat when the frequency from the base excitation is veryclose but not exactly equal to the resonant frequency of freevibrations of the cantilever system a kind of vibration calledbeating occurs and the period of the beating increases as120596 approached 120596

1 This kind of beating decays exponentially

along the time due to the mechanical dampingFigure 13 shows the experimental displacement responses

at the position of 119909 = 1204mm from transient to steady

state As the same to the theoretical results experimental dis-placement response can be regarded as the sum of waveformsfrom transient and steady states But the transient responsevanished gradually with the time due to the mechanicaldamping and only the steady response left In addition thephenomenon of beating still existed The amplitude of thisbeating phenomenon reduced gradually and the closer theexciting frequency approached to the first natural frequencyof the cantilever system the faster the amplitude of thebeating decreased

Furthermore the amplitudes of the theoreticalwaveformsin Figures 11 and 12 are different from the experimental resultin Figure 13 This may because af the following (a) The effectof mechanical damping with respect to stiffness was onlyconsidered and the mass damping was ignored in theoreticalanalysis which can also weaken the transient vibration faster(b) The first natural frequency of the cantilever system wascalculated and measured by theoretical and experimentalmethods as 815Hz and 812Hz respectively The little dif-ference made that the max amplitudes from theoretical andexperimental results did not appear at the same frequency(c) In theoretical analysis the tip-mass was equated by theinertia force and inertia moment acting at the free end ofthe cantilever so the total length for theoretical analysiswas just the length of the beam of 1204mm Howeverduring experiments the tip-displacement was measured at

12 Shock and Vibration

005

0

minus0050 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6

0 1 2 3 4 5 6 0 1 2 3 4 5 6

0 1 2 3 4 5 6 0 1 2 3 4 5 6

0 1 2 3 4 5 6

0

0

1 2 3 4 5 6

02

01

0

minus01

minus02

02

01

0

0

minus01

minus02

015

01

005

minus005

minus01

008

006

004

002

0

minus002

minus004

minus006

minus008

008

006

004

002

0

minus002

minus004

minus006

minus008

008

006

004

002

minus002

minus004

minus006

minus008

2

15

1

05

0

minus05

minus1

minus15

minus2

01

005

0

minus005

minus01Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at735

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

795

Hz (

mm

)

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at855

Hz (

mm

)

tran

sient

to st

eady

at755

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mTh

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

815

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

875

Hz (

mm

)

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at775

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

835

Hz (

mm

)

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at895

Hz (

mm

)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Figure 11 Theoretical tip displacement responses with the second-type mode shape of the beam from transient to steady state

the approximate position of 119909 = 1204mm This is justan approximate value because Eddy-Current Sensor cannotgive exact measurement results of the displacement at thejunction between the cantilever and the tip-mass This maycause that the amplitude level from theoretical discussionwashigher than from experimental measurement In additionthe amplitude of the instantaneous base displacement fromthe exciter may be not the fixed value 001mm at the startinginstant Therefore the maximum displacement responsesduring the transient state in the experimental measurementsmay shift away from the real experimental results

The theoretical responses with two improved types ofmode shapes and the experimental displacement responsesversus varying positions of beam and the frequencies atsteady state are shown in Figures 14ndash16 respectively Figures14 and 15 show two types of theoretical displacements atthe positions of 119909 = 364mm 574mm 784mm 994mmand 1204mmwith continuous varying frequencies In exper-iments the displacement responses were measured at thesame positions with theoretical calculations but with somecertain frequencies of 782Hz 792Hz 802Hz 807Hz81Hz 812Hz 817Hz 82Hz 83Hz and 84Hz as plottedin Figure 16 It is noticed that when the frequency of baseexcitation approached to the first natural frequency of thecantilever system the amplitude of forced vibration willrapidly increase and become maximum for the case when

the frequency of the excitation coincides with the naturalfrequency Table 3 lists the values of the first natural frequen-cies and the displacement responses from theoretical andexperimental results at some certain positions long the beamunder the first natural frequencies It can be seen that thetheoretical and experimental results are matched very close

Table 3 The first natural frequencies and the resonantdisplacements at some certain positions

In a word the theoretical results with two different modeshapes derived by two improved analytical methods haveextremely the same behaviors at transient and steady stateswith respect to varying frequencies from base excitationThedisplacement responses indicate close agreement betweenthe experimental and analytical results All of these resultsgive full verification for the improvement of the analyticalmethods in this paper utilized in recalculating the modeshapes of a cantilever beam when an offset distance existsbetween the gravity center of the tip-mass and the point ofattachment

5 Conclusions

This paper presents two improved analytical methods ofcalculation for natural frequencies and mode shapes of auniform cantilever beam carrying a tip-mass under an input

Shock and Vibration 13

005

0

minus005Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at735

Hz (

mm

)

0 1 2 3 4 5 6

Time (s)

0 1 2 3 4 5 6

0 1 2 3 4 5 6

02

01

0

minus01

minus02

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at795

Hz (

mm

)

0

015

01

005

minus005

minus01

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at855

Hz (

mm

)

0 1 2 3 4 5 6

Time (s)

008

006

004

002

0

minus002

minus004

minus006

minus008tran

sient

to st

eady

at755

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

m

2

15

1

05

0

minus05

minus1

minus15

minus2Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at815

Hz (

mm

)

0 1 2 3 4 5 6

008

006

004

002

0

minus002

minus004

minus006

minus008Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at875

Hz (

mm

)

0 1 2 3 4 5 6

01

005

0

minus005

minus01Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at775

Hz (

mm

)

0 1 2 3 4 5 6

Time (s)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

0 1 2 3 4 5 6

0 1 2 3 4 5 6

0

008

006

004

002

minus002

minus004

minus006

minus008Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at895

Hz (

mm

)

02

01

0

minus01

minus02

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at835

Hz (

mm

)Figure 12 Theoretical tip displacement responses with third-type mode shape of the beam from transient to steady state

Table 3 The first natural frequencies and the resonant displacements at some certain positions

Resonant frequency or positions The values of frequencies and displacement responses (Hz or mm) ErrorsTheory based on 119880(2)

1(119909) Theory based on 119880(3)

1(119909) Experiment

The 1st natural frequency 815 815 812 037119909 = 364mm 01285 01285 01277 062119909 = 574mm 03022 03022 03001 069119909 = 784mm 05279 05279 05246 063119909 = 994mm 07905 07905 07855 063119909 = 1204mm 10742 10742 10671 066

base excitation using forced vibration theory and the methodof separation of variables A simplified cantilever model byreplacing the tip-mass with an equivalent inertial force andinertial moment acting at the free end of the cantilever beamwas detailed In addition using the Dirac Delta Functionthe base transverse excitation the concentrated equivalentinertia force and inertia moment were represented in theform of distributed loads which can make the method ofseparation of variables be utilized properly In this case sometypical results of natural frequencies and mode shapes of thecantilever beam with a tip-mass were derived using the rein-forced method of separation of variables and the developedforced vibration respectively Furthermore based on modalanalysis method the transient and steady displacement

responses of the cantilever system with a tip-mass and a sinebase excitation were presented in which two different types ofmode shapes calculated by two improved analytical methodswere used Finally an experimental cantilever system wasfabricated and some practical displacement responses weremeasured

From the results of the typical natural frequencies andmode shapes we can easily found that the mode shapesfrom the first improved type were identical to the modeshapes from the second type whether the mode shapes fromthe lower order or from the higher order even the offsetdistance was large The comparing results of the transientand steady displacement responses from the theoreticalanalysis and experimental measurements further proved the

14 Shock and Vibration

005

0

minus0050 2 4 6 8 10 12

0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12

0 2 4 6 8 10 120 2 4 6 8 10 12

0 2 4 6 8 10 12 0 2 4 6 8 10 120 2 4 6 8 10 12

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at732

Hz (

mm

)

0

008

006

004

002

minus002

minus004

minus006

minus008

0

008

006

004

002

minus002

minus004

minus006

minus008

0

008

006

004

002

minus002

minus004

minus006

minus008

0

015

01

005

minus005

minus01

01

005

0

minus005

minus01

2

15

1

05

0

minus05

minus1

minus15

minus2

03

02

01

0

minus01

minus02

03

02

01

0

minus01

minus02

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at752

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at772

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at792

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at812

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at832

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at852

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at872

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at892

Hz (

mm

)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Figure 13 Experimental displacement responses at the position of 119909 = 1204mm from transient to steady state

12

1

08

06

04

02

0120 100 80 60 40 20

78

80

82

84Theo

retic

al st

eady

disp

lace

men

t (m

m)

Position at the beam (mm)Freq

uency

(Hz)

Figure 14 Theoretical steady-displacement responses versus posi-tions and frequencies with mode 119880(2)

1(119909)

improvements of the analytical methods for calculating themode shapes of the cantilever beam with a tip-mass anda base excitation In addition the amplitudes of the modeshapes decreased with the increasing mass value of the tip-mass when the moment of inertia and offset were constantsThe amplitude of the first-order mode shape increased but

12

1

08

06

04

02

0120 100 80 60 40 20

78

80

82

84Theo

retic

al st

eady

disp

lace

men

t (m

m)

Position at the beam (mm)Freq

uency

(Hz)

Figure 15 Theoretical steady-displacement responses versus posi-tions and frequencies with mode 119880(3)

1(119909)

the other nine mode shapes decreased with the increasingmoment of inertia when the mass and offset were set asconstants Conversely the amplitude of the first mode shapedecreased but the other ninemode shapes increased with theincreasing offset distance under the condition that the massand moment of inertia were constants Globally the effect

Shock and Vibration 15

12

1

08

06

04

02

0

Expe

rimen

tal s

tead

y di

spla

cem

ent (

mm

)

120 100 80 60 40 20Position at the beam (mm)

78

80

82

84Freq

uency

(Hz)

Figure 16 Experimental steady-displacement responses versuspositions and frequencies

from varying combinations of the mass and moment inertiaof the tip-mass and the offset distance for the lower-ordermode shapes was stronger than for the higher ones

Furthermore it is clearly noticed that the mass of tip-mass can reduce the values of the natural frequencies andthe increasing mass value gave the decreasing frequencieswhen the moment of inertia of the tip-mass and the offsetdistance were set as constants Similarly the moment ofinertia can further decrease the natural frequencies and thehigher moment of inertia resulted in the lower frequencieswhen the mass and the offset were constants Differently theoffset distance can reduce the first-order natural frequenciesbut increase the other nine order frequencies and the higheroffset distance led to the lower first-order natural frequenciesbut produced the higher other nine order frequencies whenthe mass and the moment of inertia were set to constant Inglobal view the influence from the varying mass momentof inertia and offset distance for the lower-order naturalfrequencies was larger than for the higher-order ones weatherthis influence was negative or positive

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the Natural Science Foundationsof China (no 51275380) and the Scientific and TechnologicalProject of Shaanxi Province (no 2012K06-36) The authorswould like to thank the reviewers for their valuable commentson this paper

References

[1] P A A Laura J L Pombo and E A Susemihl ldquoA note on thevibrations of a clamped-free beam with a mass at the free endrdquoJournal of Sound and Vibration vol 37 no 2 pp 161ndash168 1974

[2] B Rama Bhat andHWagner ldquoNatural frequencies of a uniformcantilever with a tip mass slender in the axial directionrdquo Journalof Sound and Vibration vol 45 no 2 pp 304ndash307 1976

[3] C W S To ldquoVibration of a cantilever beam with a baseexcitation and tip massrdquo Journal of Sound and Vibration vol83 no 4 pp 445ndash460 1982

[4] P A A Laura ldquoComments on lsquoVibration of a cantilever beamwith a base excitation and tip massrsquordquo Journal of Sound andVibration vol 88 no 4 p 569 1983

[5] C W S To ldquoDiscussion on lsquovibration of a cantilever beam witha base excitation and tip massrsquordquo Journal of Sound and Vibrationvol 97 no 2 pp 349ndash351 1984

[6] R G Jacquot ldquoFurther comments on lsquoVibration of a cantileverbeam with a base excitation and tip massrsquordquo Journal of Sound andVibration vol 93 no 2 pp 312ndash313 1984

[7] S AMousavi Lajimi andG R Heppler ldquoComments on lsquonaturalfrequencies of a uniform cantilever with a tip mass slender inthe axial directionrsquordquo Journal of Sound and Vibration vol 331 no12 pp 2964ndash2968 2012

[8] R B Bhat ldquoAuthors response to the comments on lsquonaturalfrequencies of a uniform cantilever with a tip mass slender inthe axial directionrsquordquo Journal of Sound and Vibration vol 331 no12 pp 2962ndash2963 2012

[9] E Esmailzadeh and G Nakhaie-Jazar ldquoPeriodic behavior ofa cantilever beam with end mass subjected to harmonic baseexcitationrdquo International Journal of Non-Linear Mechanics vol33 no 4 pp 567ndash577 1998

[10] H Abramovich and O Hamburger ldquoVibration of a cantilevertimoshenko beam with a tip massrdquo Journal of Sound andVibration vol 148 no 1 pp 162ndash170 1991

[11] B Pratiher ldquoVibration control of a transversely excited can-tilever beam with tip massrdquo Archive of Applied Mechanics vol82 no 1 pp 31ndash42 2012

[12] M Kim M Hoegen J Dugundji and B L Wardle ldquoModelingand experimental verification of proof mass effects on vibrationenergy harvester performancerdquo Smart Materials and Structuresvol 19 Article ID 045023 2010

[13] Y T Li and R Wong ldquoIntegral and series representations ofthe Dirac delta functionrdquo Communications on Pure and AppliedAnalysis vol 7 no 2 pp 229ndash247 2008

[14] M Al-Gwaiz Sturm-LiouvilleTheory and its Applications chap-ter 3 Springer London UK 2008

[15] F Bao-lian ldquoGeneralized reciprocal theorems and their appli-cationsrdquo Applied Mathematics and Mechanics vol 23 no 2 pp203ndash210 2001

[16] C Cai H Zheng M Khan and K HungModeling of MaterialDamping Properties in ANSYS Defense Systems Division Insti-tute of High Performance Computing 2002

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4 Shock and Vibration

at 119909 = 119871

119864119868

1198893

1198891199093119906119886(119909 119905) = [119872

1205972119906119886(119909 119905)

1205971199052

+ 119903119888119872

1205972

1205971199052(

120597119906119886(119909 119905)

120597119909

)]

(11)

119864119868

1198892

1198891199092119906119886(119909 119905)

= minus [119903119888119872

1205972119906119886(119909 119905)

1205971199052

+ (119869 + 1199032

119888119872)

1205972

1205971199052(

120597119906119886(119909 119905)

120597119909

)]

(12)

The solutions of (8) subjected to four boundary condi-tions and two initial conditions can be obtained convenientlyby the traditional method of separation of variables Themethod regards the response as a superposition of the systemeigen-functionsmultiplied by corresponding time dependentgeneralized coordinates Of course this necessitates firstobtaining the solution of the system eigen-value problemTurning attentions to the corresponding eigen-value problemand considering the free vibration characterized by setting119887(119905) = 0 the solution of (8) becomes separable in space andtime Letting

119906119886(119909 119905) = 119880 (119909) 119879 (119905) (13)

and substituting (13) into the free vibration form of (8) twodependent equations are obtained using the separation ofvariables method as

119889

1198891199052119879 (119905) + 120596

2119879 (119905) = 0 (14)

1198894

1198891199094119880 (119909) minus 120573

4119880 (119909) = 0 (15)

where 1205962 indicates the angular frequency and 1205734 =

1198981205962(119864119868) In this case the four boundary conditions will be

changed as

at 119909 = 0

119880 (119909) = 0 (16)

119889

119889119909

119880 (119909) = 0 (17)

at 119909 = 119871

119864119868

1198893

1198891199093119880 (119909) = minus [120596

2119872119880(119909) + 119903

1198881198721205962(

119889

119889119909

119880 (119909))] (18)

119864119868

1198892

1198891199092119880 (119909) = 119903

1198881198721205962119880 (119909) + (119869 + 119903

2

119888119872)1205962(

119889

119889119909

119880 (119909))

(19)

The general solution of (15) can be easily derived to be

119880 (119909) = 1198621cos (120573119909) + 119862

2sin (120573119909) + 119862

3cosh (120573119909)

+ 1198624sinh (120573119909)

(20)

where 119862119899(119899 = 1 2 3 4) are constants of integration In

order to evaluate three of these constants in terms of thefourth and derive the characteristic equation the boundaryconditions (16) to (19) must be used Indeed substituting (20)into (16) through (19) gives the characteristic equation andthe mode shapes as

1 + 119888119888ℎ minus 120601119906 (119904119888ℎ minus 119888119904ℎ)

minus 1198771[119904119888ℎ + 119888119904ℎ minus 120601119906 (1 minus 119888119888ℎ)]

minus 21198772119904119904ℎ minus 119877

2

2(1 minus 119888119888ℎ) = 0

(21)

119880 (119909) = 1198622sin (120573119909) minus 119904ℎ (120573119909) minus 120582 [cos (120573119909) minus 119888ℎ (120573119909)]

(22)

where 120582 = 1198601211986011and the constant 119862

2is an arbitrary value

and in which

11986011= minus(1 + 119877

2)119888 + (119877

2minus 1)119888ℎ + 119877

1(119904 + 119904ℎ) 119860

12=

1198771(119888ℎ minus 119888) minus (1 + 119877

2)119904 + (119877

2minus 1)119904ℎ

119906 = 120573119871 119904 = sin(119906) 119888 = cos(119906) 119904ℎ = sinh(119906) 119888ℎ =cosh(119906)

120601 = 119872(119898119871) 1198771= 120601119906

3[(119870119871)

2+ (119903119888119871)2] 1198772=

1206011199062(119903119888119871) 119870 = radic119869119872

23TheDevelopedMethod of Forced Vibration If we treat thebase displacement excitation as a kind of boundary conditionthen the differential equation of motion and four boundaryconditions of the cantilever system can be written as

119864119868

1205974119906119886(119909 119905)

120597119909

+ 119898

1205972119906119886(119909 119905)

120597119905

= 0 (23)

at 119909 = 0

119906119886(0 119905) = 119887 (119905) (24)

119889

119889119909

119906119886(0 119905) = 0 (25)

at 119909 = 119871

119864119868

1198893

1198891199093119906119886(119909 119905)

= [119872

1205972119906119886(119909 119905)

1205971199052

+ 119903119888119872

1205972

1205971199052(

120597119906119886(119909 119905)

120597119909

)]

(26)

119864119868

1198892

1198891199092119906119886(119909 119905)

= minus [119903119888119872

1205972119906119886(119909 119905)

1205971199052

+ (119869 + 1199032

119888119872)

1205972

1205971199052(

120597119906119886(119909 119905)

120597119909

)]

(27)

Shock and Vibration 5

Under the simple harmonic excitation 119890119895120596119905 the steady statesolution of (23) can be regarded with the form as

119906119886(119909 119905) = 119880

119886(119909 120596) 119890

119895120596119905minus120593

= [1198611cos (120573119909) + 119861

2sin (120573119909)] 119890119895120596119905minus120593

+1198613cosh (120573119909) + 119861

4sinh (120573119909)]

(28)

where 119861119899(119899 = 1 2 3 4) are arbitrary constants the

parameter 120573 is defined by 1205734 = 1198981205962(119864119868) 119895 is the imaginaryunit and 120593 is the lagging phase

We recalculated the natural frequencies and the modeshapes directly by substituting (28) into four boundaryconditions (24) through (27) and obtained the characteristicequation and displacement response which are different from[3]

119863 = 1 + 119888119888ℎ minus 120601119906 (119904119888ℎ minus 119888119904ℎ)

minus 1198771[119904119888ℎ + 119888119904ℎ minus 120601119906 (1 minus 119888119888ℎ)]

minus 21198772119904119904ℎ minus 119877

2

2(1 minus 119888119888ℎ)

119880 (119909) = (

119861

2119863

) 1198651015840

1119906cos (120573119909) + 1198651015840

2119906cosh (120573119909)

+1198651015840

3119906[sin (120573119909) minus sinh (120573119909)]

(29)

where 119861 is defined by 119887(119905) = 119861119890119895120596119905 and in which

1198651015840

1119906= 1+119888119888ℎminus119904119904ℎminus2120593119906119904119888ℎminus2119877

1119888119904ℎ+(119877

1120593119906minus119877

2

2)(1minus

119888119888ℎ + 119904119904ℎ) minus 21198772(119888119888ℎ + 119904119904ℎ) =119865

1119906minus 21198772(119888119888ℎ + 119904119904ℎ)

1198651015840

2119906= 1+119888119888ℎ+119904119904ℎ+2120593119906119888119904ℎminus2119877

1119904119888ℎ+(119877

1120593119906minus119877

2

2)(1minus

119888119888ℎ minus 119904119904ℎ) + 21198772(119888119888ℎ minus 119904119904ℎ) =119865

2119906+ 21198772(119888119888ℎ minus 119904119904ℎ)

1198651015840

3119906= (119904119888ℎ + 119888119904ℎ)(1 minus 119877

1120593119906 +119877

2

2) + 2[120593119906119888119888ℎ minus 119877

1119904119904ℎ minus

1198772(119904119888ℎ minus 119888119904ℎ)] = 119865

3119906

It is obvious that the characteristic equation derived by thereinforced method of separation of variables is equal tothe equation obtained by the developed method of forcedvibration

Themode shapes cannot be obtained directly because theboundary conditions contain the parameter1205962 However thisproblem can be successfully solved by Dirac Delta Function[13] Thus the mode shapes may be obtained by the Sturm-Liouville theory [14] and expressed as

119880119894(119909) = (

1

2

) 1198651015840

1119906119894119865 cos (120573

119894119909) + 119865

1015840

2119906119894cosh (120573

119894119909)

+ 1198651015840

3119906119894[sin (120573

119894119909) minus sinh (120573

119894119909)]

(30)

3 Computed Results and Comparisons

In this section three types of the natural mode shapes ofthe cantilever beam with a tip-mass were computed and

compared The first one is called the undeveloped modeshapes which is directly obtained from [3] by the expression

119880(1)

119894(119909) = (

1

2

) 1198651119894119888

cos(119906119894119909

119871

) + 1198652119894119888

cosh (119906119894119909

119871

)

+ 1198653119894119888[sin(

119906119894119909

119871

) minus sinh(119906119894119909

119871

)]

(31)

where 1198651119894119888

= 1198651119894119888ℎ(119906119894) 1198652119894119888

= 1198652119894119888ℎ(119906119894) and 119865

3119894119888=

1198653119894119888ℎ(119906119894) in which 119865

1119894 1198652119894 and 119865

3119894are the functions 119865

1119906 1198652119906

and 1198653119906

with the variable 119906 replaced by 119906119894 respectively

The second one is based on the expression of thedeveloped mode shapes derived in this paper Let (30) bemultiplied by 21198651015840

3119894to give the second mode shapes as

119880(2)

119894(119909) = sin(

119906119894119909

119871

) minus sinh(119906119894119909

119871

) + (

1198651015840

1119894

1198651015840

3119894

) cos(119906119894119909

119871

)

+ (

1198651015840

2119894

1198651015840

3119894

) cosh (119906119894119909

119871

)

(32)

At last the arbitrary constant 1198622in (22) was chosen as

constant 1 to give the third type of the natural mode shapes as

119880(3)

119894(119909) = sin(

119906119894119909

119871

) minus 119904ℎ (

119906119894119909

119871

)

minus 120582 [cos(119906119894119909

119871

) minus 119888ℎ (

119906119894119909

119871

)]

(33)

Before the mode shapes are determined the roots ofthe characteristic equation must be calculated firstly Thecharacteristic equation (21) or 119863 = 0 can be solved by twosteps First approximate roots were pointed out by applyinggraphing method Then the approximate roots were iteratedby the bisection method to give the exact characteristic rootswith given error All these processes were finished usingMATLAB software

Because the condition that the gravity center of the tip-mass coincided with the point of the end attachment hasbeen verified in [6] we focus on the condition when theoffset distance is not zero Table 1 shows the first five roots ofthe characteristic equations from five samples with differentvalues of 120601 119896119871 and 119889119871 It can be clearly noticed thatthe larger value of 119889119871 resulted in the less value of the firstcharacteristic root but the larger values of the other fourcharacteristic roots when 120601 and 119896119871 were set as constants

Once the roots of the characteristic equation have beenobtained three types of the first five mode shapes of thecantilever structure from the five given samples can becalculated Figures 4 5 6 7 and 8 show the first five modeshapes of the cantilever with various combinations of 120601 119896119871and 119889119871

6 Shock and Vibration

Table 1 Typical roots of characteristic equation with variables 120601 119896119871 and 119889119871

Sample 120601 119870119871 119889119871

Roots of characteristic equation1199061

1199062

1199063

1199064

1199065

1 4 4 0 035211 129234 477818 788401 11017802 4 4 05 034814 130637 477952 788472 11018273 4 4 1 034200 132825 478225 788634 11019404 4 4 2 032544 138983 479177 789233 11023685 4 4 4 028806 154543 482543 791455 1103990

003002001

0

0 05 1 0 05 1 0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

000000

050000

100000

i = 1

000000

050000

100000

i = 1

000000 000000

000000

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

000000

050000

100000

i = 2

000000

050000

100000

i = 2

minus186571 times 10minus9

000792

003152

minus198560 times 10minus8

minus9418658

minus18545378

0

minus100

minus2000 05 1

0 05 1

0 05 1

0 05 1

1050

times104 i = 3

000000

050000

100000

i = 3

000000

050000

100000

i = 3

786191 times 10minus6

111026 times 104

minus686262 times 103

50

minus5

times105 i = 4000000

050000

100000

i = 4

000000

050000

100000

i = 4

i = 5000000

050000

100000

i = 5

000000 000000

050000

100000

i = 5times106

20

minus2

000017

minus271300 times 106

minus855944 times 104

1

05

0

1

05

0

03

02

01

0

03

02

01

0

151

050

151

050

2

0

minus2

2

0

minus2

210

minus1

210

minus1

minus493414 times 10minus8

032943

131050

367824 times 10minus11

017448

034354

113682 times 10minus10

160541

minus009923

457130 times 10minus11

minus002114

006144

866744 times 10minus11

minus140614

minus004436

032943

131050

017448

034354

160541

minus009923

000000

minus002114

006144

minus140614

minus004436

Notes at xL Notes at xL Notes at xLValues ValuesValues

Ui(x)

xL

Ui(x)

xL

Ui(x)

xL

minus233509 times 10minus5

107963 times 104

minus313859 times 104

First-type mode shapes U(1)i

(x) Second-type mode shapes U(2)i

(x) Third-type mode shapes U(3)i

(x)

Figure 4 Three types of the first five mode shapes with the combinations of 120601 = 4 119896119871 = 4 and 119889119871 = 0

It is obvious that when 119889119871 = 0 the first type of naturemode shapes 119880(1)

119894(119909) derived by the undeveloped method

turned far away from the third-type mode shapes 119880(3)119894(119909)

derived by the reinforced method of separation of variablesIn addition the larger value of 119889119871 resulted in the moreinconsistent mode shapes However the second type of modeshapes 119880(2)

119894(119909) obtained by the developed method of forced

vibration seemed in good agreement with the third onesFurthermore the amplitude of the first mode shape

decreased but the other four mode shapes increased with theincreasing value of the 119889119871 when 119896119871 and 120601 were fixed to be

constants The effect from the varying value of 119889119871 for thelower mode shapes was always stronger than for the higher

4 Experimental Verification

In order to experimentally explain and compare two typesof natural mode shapes derived by two improved methodsabove the transient and steady responses of the cantileversystem under an input base excitation should be calculatedfirst This whole calculation can be finished by method ofmodal analysis But above all the orthogonality property of

Shock and Vibration 7

0 05 1

0 05 1

Ui(x)

xL

Ui(x)

xL

Ui(x)

xL

014013012011

0

minus100

minus200

000000

050000

100000

i = 1

000000

050000

100000

i = 2

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

011044

011840

014143

533123

minus9287788

minus18666064

0 05 1

1050

times104

000000

050000

100000

i = 3minus8449356

107519 times 105

minus970528 times 103

0 05 1

50

minus5

times105

000000

050000

100000

i = 422493344

139452 times 104

139108 times 105

0 05 1

40

minus4minus8

times106

000000

050000

100000

i = 5minus42806116

minus266045 times 106

minus765307 times 106

0 05 1

000000

050000

100000

i = 106

04

02

0

392120 times 10minus9

017490

069258

0 05 1

06

04

02

0

0 05 1

000000

050000

100000

i = 2547973 times 10minus11

017881

034942

03

02

01

00 05 1

03

02

01

0

0 05 1

000000

050000

100000

i = 3minus505260 times 10minus11

160517

minus010133

151

050

0 05 1

151

050

0 05 1

000000

050000

100000

i = 4minus776048 times 10minus11

minus002162

006261

2

0

minus20 05 1

2

0

minus2

0 05 1

000000

050000

100000

i = 5663249 times 10minus11

minus140614

minus004516

210

minus1

000000

000000

000000

000000

017490

069258

017881

034942

160517

minus010133

minus002162

006261

000000

050000

100000

i = 5

0 05 1

210

minus1

000000

minus140614

minus004516

Notes at xL Notes at xL Notes at xLValues ValuesValuesFirst-type mode shapes U(1)i

(x) Second-type mode shapes U(2)i

(x) Third-type mode shapes U(3)i

(x)

Figure 5 Three types of the first five mode shapes with the combinations of 120601 = 4 119896119871 = 4 and 119889119871 = 05

034

033

032

031

0 05 1 0 05 1 0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1 0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0

minus100

minus200

10

5

0

times104

50

minus5

times105

times106

0minus5minus10minus15

Ui(x)

xL

Ui(x)

xL

Ui(x)

xL

04

02

0

04

02

0

04

02

0

04

02

0

151

050

151

050

2

0

minus2

2

0

minus2

210

minus1

210

minus1

000000

050000

100000

i = 1

000000 000000

050000

100000

i = 1

030899

031712

033934

000000

050000

100000

i = 2861965

minus9105604

minus18440224

000000

050000

100000

i = 3minus10637851

973291 times 104

minus123754 times 104

000000

050000

100000

i = 419388076

168672 times 104

311790 times 105

000000

050000

100000

i = 5minus19120922

minus244086 times 106

minus152554 times 107

000000

050000

100000

i = 1

331605 times 10minus9

012125

047836

000000

050000

100000

i = 2

minus550109 times 10minus11

018658

036219

000000

050000

100000

i = 3

minus175471 times 10minus11

160459

minus010631

000000

050000

100000

i = 4

109777 times 10minus10

minus002275

006562

000000

050000

100000

i = 5

836395 times 10minus11

minus140614

minus004731

012125

047836

000000 000000

050000

100000

i = 2

018658

036219

000000 000000

050000

100000

i = 4

minus002275

006562

000000 000000

050000

100000

i = 5

minus140614

minus004731

000000 000000

050000

100000

i = 3

160459

minus010631

Notes at xL Notes at xL Notes at xLValues ValuesValuesFirst-type mode shapes U(1)i

(x) Second-type mode shapes U(2)i

(x) Third-type mode shapes U(3)i

(x)

Figure 6 Three types of the first five mode shapes with the combinations of 120601 = 4 119896119871 = 4 and 119889119871 = 1

8 Shock and Vibration

088087086085

0 05 1 0 05 1 0 05 1

0 05 1 0 05 1

0 05 10 05 1

0 05 1 0 05 1

0 05 1 0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0

minus50

minus100

minus150

6420

minus2

times104

840

minus4

times105

0minus1minus2minus3

times107

0302010

03

02

01

0

04

02

0

04

02

0

151

050

151

050

210

minus1

210

minus1

210

minus1

210

minus1

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

084906

085736

087731

minus1308377

minus9308045

minus16072843

63490847

565642 times 104

minus172997 times 104

minus329762092

230021 times 104

667852 times 105

946242416

minus149912 times 106

minus306922 times 107

minus995048 times 10minus10

007811

030669

511030 times 10minus11

021118

040600

minus114494 times 10minus10

160244

minus012466

688252 times 10minus11

minus002686

007708

183045 times 10minus11

minus140613

minus005561

000000

000000

000000

000000

000000

007811

030669

021118

040600

160244

minus012466

minus002686

007708

minus140613

minus005561

Ui(x)

xL

Ui(x)

xL

Ui(x)

xL

Notes at xL Notes at xL Notes at xLValues ValuesValuesFirst-type mode shapes U(1)i

(x) Second-type mode shapes U(2)i

(x) Third-type mode shapes U(3)i

(x)

Figure 7 Three types of the first five mode shapes with the combinations of 120601 = 4 119896119871 = 4 and 119889119871 = 2

the natural mode shapes should be derived In fact usingBetti theorem [15] the orthogonality properties with respectto mass and stiffness can be obtained respectively as follows

int

119871

0

119898119880(119896)

119903(119909)119880(119896)

119894(119909) 119889119909 +119872119880

(119896)

119903(119871)119880(119896)

119894(119871)

+ 119903119888119872119880(119896)

119894(119871)

119889119880(119896)

119903(119871)

119889119909

+ 119903119888119872119880(119896)

119903(119871)

119889119880(119896)

119894(119871)

119889119909

+ (119869 + 1199032

119888119872)

119889119880(119896)

119894(119871)

119889119909

119889119880(119896)

119903(119871)

119889119909

= (120575(119896)

119903119894)

2

(119896 = 1 2 119903 119894 = 1 2 )

(34)

int

119871

0

119864119868

1198892119880(119896)

119894(119909)

1198891199092

1198892119880(119896)

119903(119909)

1198891199092

119889119909 = (120596119903120575(119896)

119903119894)

2

(119896 = 1 2 119903 119894 = 1 2 )

(35)

In this case the normalized mode shapes can be defined as

119880(119896)

119903119873(119909) =

119880(119896)

119903(119909)

120575(119896)

119903119903

(36)

where

(120575(119896)

119903119903)

2

= int

119871

0

119898[119880(119896)

119903(119909)]

2

119889119909 +119872[119880(119896)

119903(119871)]

2

+ 2119903119888119872119880(119896)

119903(119871) [

119889

119889119909

119880(119896)

119903(119871)]

+ (119869 +1198721199032

119888) [

119889

119889119909

119880(119896)

119903(119871)]

2

(37)

41 Theoretical Responses of the Cantilever System with a BaseSine Excitation According to the method of modal analysisthe solution of (8) has the form of

119906119886(119909 119905) =

infin

sum

119903=1

119880(119896)

119903119873(119909) 119902119903(119905) (38)

So inserting (38) into (8) gives

119864119868

infin

sum

119903=1

1198894119880(119896)

119903119873(119909)

1198891199094

119902119903(119905) + 119898

infin

sum

119903=1

119880(119896)

119903119873(119909)

1198892119902119903(119905)

1198891199052

= [119872120575 (119909 minus 119871) + 119903119888119872

119889120575 (119909 minus 119871)

119889119909

minus 119898]

1198892119887 (119905)

1198891199052

(39)

Considering the orthogonality properties of equations (34)and (35) let (39) be multiplied by119880(119896)

119894119873(119909) and integrated over

Shock and Vibration 9

192

191

19

0 05 1 0 05 1

0 05 1 0 05 1

0 05 1

0 05 10 05 1

0 05 10 05 1

0 05 1 0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0

minus200

minus400

minus600

0minus5minus10

times104

1

0

minus1

times106

times107

0minus2minus4minus6

02

01

0

02

01

0

06

04

02

0

06

04

02

0

15

1

05

0

15

1

05

0

210

minus1

210

minus1

210

minus1

210

minus1

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

000000

050000

100000

i = 1

000000

050000

100000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

189913

190621

192051

minus56420429

minus28138349

4723398

134513 times 104

minus110859 times 105

minus260952 times 104

minus615688 times 104

496921 times 104

145091 times 106

168417 times 105

241806 times 106

minus633899 times 107

137503 times 10minus9

004964

019458

minus238241 times 10minus11

028530

054402

159443

minus019150

minus283947 times 10minus11

minus004212

012050

771494 times 10minus11

minus140605

minus008755

000000

000000

000000 000000

000000

000000

004964

019458

028530

054402

159443

minus019150

minus004212

012050

minus140605

minus008755

Ui(x)

xL

Ui(x)

xL

Ui(x)

xL

Notes at xL Notes at xL Notes at xLValues ValuesValues

minus387890 times 10minus11

First-type mode shapes U(1)i

(x) Second-type mode shapes U(2)i

(x) Third-type mode shapes U(3)i

(x)

Figure 8 Three types of the first five mode shapes with the combinations of 120601 = 4 119896119871 = 4 and 119889119871 = 4

the domain 0 lt 119909 lt 119871 to give a set of independent ordinarydifferential equations as

1198892119902119894(119905)

1198891199052

+ 1205962

119894119902119894(119905) = 119891

(119896)

119894(119905) (40)

Considering the properties of Dirac Delta Function

int

119871

0

119880(119896)

119894119873(119909) 120575 (119909 minus 119871) 119889119909 = 119880

(119896)

119894119873(119871)

int

119871

0

119880(119896)

119894119873(119909)

119889120575 (119909 minus 119871)

119889119909

119889119909 = minus

119889119880(119896)

119894119873(119871)

119889119909

(41)

the generalized forces associated with the generalized coordi-nates 119902

119894(119905) can be obtained as

119891(119896)

119894(119905)

= [119872119880(119896)

119894119873(119871) minus 119903

119888119872

119889119880(119896)

119894119873(119871)

119889119909

minus 119898int

119871

0

119880(119896)

119894119873(119909) 119889119909]

1198892119887 (119905)

1198891199052

(42)

Furthermore considering the mechanical damping asRayleigh damping and ignoring the part of mass dampingthe damping coefficient 120577 of the bimorph beam can be deter-mined by the modal damping ratio which can be obtained

by experiment [16] The independent ordinary differentialequations with damping then can be written as

1198892119902119894(119905)

1198891199052

+ 2120596119894120577

119889119902119894(119905)

119889119905

+ 1205962

119894119902119894(119905)

= [119872119880(119896)

119894119873(119871) minus 119903

119888119872

119889119880(119896)

119894119873(119871)

119889119909

minus 119898int

119871

0

119880(119896)

119894119873(119909) 119889119909]

1198892119887 (119905)

1198891199052

(43)

Now (43) can be solved using Laplace transform Applyinga sine excitation 119887(119905) = 119861 sin(120596119905) to the base of cantileverstructure under zero initial conditions the transfer functionof the independent differential equations for the cantileverbeam with tip-mass under base displacement excitation canbe written as

119866119894(119904)

=

119876119894(119904)

119861119894(119904)

=

(minus1205962)[119872119880

(119896)

119894119873(119871)minus119903

119888119872(119889119880

(119896)

119894119873(119871) 119889119909)minus119898int

119871

0119880(119896)

119894119873(119909) 119889119909]

1199042+ 2120596119894120577119904 + 120596

2

119894

(44)

10 Shock and Vibration

where 119876119894(119904) and 119861

119894(119904) are the Laplace transforms of 119902

119894(119905) and

119887(119905) respectively The inverse transformation of the transferfunction will be further obtained as

119892119894(119905)

=

(minus1205962)[119872119880

(119896)

119894119873(119871)minus119903

119888119872(119889119880

(119896)

119894119873(119871) 119889119909)minus119898int

119871

0119880(119896)

119894119873(119909) 119889119909]

120596119894radic1 minus 120577

2

times 119890minus120596119894120577119905 sin(120596

119894radic1 minus 120577

2119905)

(45)

In this case the solutions of (43) can be solved by convolutiontheorem as

119902119894(119905) = 119887 (119905) lowast 119892

119894(119905) = int

119905

0

119887 (120591) 119892119894(119905 minus 120591) 119889120591 (46)

Finally substituting the result of (46) into (38) and after somelengthy algebraic manipulation the absolute displacementresponse of the cantilever beam with tip-mass under basetransverse excitation can be obtained as

119906119886(119909 120596 119905) =

infin

sum

119894=1

119880(119896)

119894119873(119909)

119861 [119872119880(119896)

119894119873(119871) minus 119903

119888119872(119889119880

(119896)

119894119873(119871) 119889119909) minus 119898int

119871

0119880(119896)

119894119873(119909) 119889119909]

radic1205964+ 2 (2120577

2minus 1) 120596

21205962

119894+ 1205964

119894

times [

1205963

120596119894radic1 minus 120577

2

119890minus120596119894120577119905 sin(120596

119894radic1 minus 120577

2119905 + 1206011) + 1205962 sin (120596119905 + 120601

2)]

(47)

where 1206011and 120601

2are determined by tan(120601

1) = 2120577radic1 minus 120577

2

((120596120596119894)2+ 21205772minus 1) and tan(120601

2) = minus2120577(120596120596

119894)(1 minus (120596120596

119894)2)

It can be seen that the solution consists of two parts abouttimeThe first term represents free vibration which will decaywith time exponentially The second term depending on theexcitation represents the forced vibration of the system

42 Experimental Responses and Comparisons with the Theo-retical Results In order to explain the absolute displacementresponse of the cantilever structure under base excitationexperimentally a typical cantilever beam in which theelongated beam was inserted and fixed into the tip-mass tomake sure the gravity center of tip-mass was collinear withthe beam was fabricated (as shown in Figure 9) The chosengeometric and physical properties of the cantilever beamincluding the tip-mass are given in Table 2

The complete experimental devices are shown inFigure 10 The electromagnetic vibrating table type SW-TFAcontrolled by a programmable controller type SW-F-2TF-2was used to create a specific harmonic input excitationThe cantilever beam with a tip-mass was clamped onthe vibrating top platform Eddy-Current Sensors typeDM6008 and ZA210800 were used to measure the inputdisplacement from base vibrating exciter and the absolutedynamic displacement from the beam respectively Allmeasurements from cantilever beam and vibrating exciterwere connected to Digital Oscilloscope type DS1204Band the Industrial Computer including a PC-based DataAcquisitionModule from ADLINK company type DAQ2213The Digital Oscilloscope and Industrial Computer displayedand processed the measurement results using the DAQPilotsoftware and the results from subsequent analysis using theMATLAB software

Table 2 Characteristic properties of the cantilever beam includingtip-mass

Physical properties Beam Tip-massYoungrsquos modulus 119864 (GPa) 69 mdashDensity 120588

119904or 120588119898(kgm3) 2630 7753

Length 119871 or 119871119898(mm) 1204 156

Thickness 119905119904or 119905119898(mm) 49 295

Width 119908 (mm) 102 102Total mass119872 (g)a mdash 3383Offset 119903

119888(mm)b mdash 801

Rotary inertia 119869 (kgsdotm2)b c mdash 333 times 10minus6aThe total mass of the tip-mass including the elongated part of beam wascalculated by119872 = 119871

119898119905119898119908120588119898+ 119871119904119905119904119908(120588119904minus 120588119898) where 119871

119904= 10mm

bThe offset distance and the rotary inertia were calculated according to thegeometry and material properties of the beam and tip-mass using equations119903119888= (120588

1198981199051198981198712

119898minus 1205881198981199051199041198712

119904+ 1205881199041199051199041198712

119904)(2[120588

119898119905119898119871119898+ 119871119904119905119904(120588119904minus 120588119898)])

119869 = 120588119898119871119898119905119898119908 [(112)(119871

2

119898+ 1199052

119898) + ((12)119871

119898minus 119903119888)2] + 119871

119904119905119904119908(120588119904minus

120588119898)[(112)(119871

2

119904+ 1199052

119904) + (119903119888minus (12)119871

119904)2]

cThe corresponding parameters 120601 119896119871 and 119903119888119871 were calculated as follows

120601 = 119872(120588119904119905119904119908) = 21377 119896119871 = radic119869119872119871 = 00824 and 119903

119888119871 = 00665

The analytical model defined by (47) in which two typesof mode shapes derived by two improved analytical methodsare utilized can be used to analyze the steady and transientdisplacement responses of the cantilever beam with a tip-mass and an input base excitation based on the chosenvarying frequencies time and positions along the lengthof the beam Typically the displacement responses frompositions along 119909-axis at 119909

1= 364mm 119909

2= 574mm

1199093

= 784mm 1199094

= 994mm and 1199095

= 1204mmoff the origin point (as shown in Figure 9) were measured

Shock and Vibration 11

y

b(t)

x1

x2

x3

x4

x5

Ls

L

t s

t m x

Lm

rc

Cg

o998400

Figure 9 Typical cantilever beam with a tip-mass

1 2

3

56

7

8

4

Tip-massCantilever beam

(7) Cantilever with a tip-mass

(1) Industrial computer including data acquisitionmodule type DAQ2213

(2) Digital oscilloscope type DS1204B(3) Programmable controller type SW-F-FTF-2(4) Power supply type DP1308A(5) Driver modules for Eddy-current sensors(6) Electromagnetic vibrating table type SW-TFA

(8) Eddy-current sensors type DM6008 and ZA210800

Figure 10 Complete experimental setup

by Eddy-Current Sensors The amplitude of the input sine-displacement excitation for analysis was measured as 119861 =

001mm and the damping ratio for the transverse vibrationaround the fundamental resonant frequency was found to be120577 = 00025 by Logarithmic Decrement Method

In the following paragraphs we focus on the transientand steady responses of the cantilever system with a sine baseexcitation

Figures 11 and 12 show the theoretical displacementresponses at the free end of the beam from transient tosteady state with two improved mode shapes above at thefirst resonant frequency It can be seen obviously that thedisplacement responses with two improved mode shapeswe proposed are almost identical This richly proves thatthe first-type mode shapes reformulated by the reinforcedmethod of separation of variables are equal to the second-type mode shapes derived by the developed method offorced vibration In addition it can be further obtainedthat when the frequency from the base excitation is veryclose but not exactly equal to the resonant frequency of freevibrations of the cantilever system a kind of vibration calledbeating occurs and the period of the beating increases as120596 approached 120596

1 This kind of beating decays exponentially

along the time due to the mechanical dampingFigure 13 shows the experimental displacement responses

at the position of 119909 = 1204mm from transient to steady

state As the same to the theoretical results experimental dis-placement response can be regarded as the sum of waveformsfrom transient and steady states But the transient responsevanished gradually with the time due to the mechanicaldamping and only the steady response left In addition thephenomenon of beating still existed The amplitude of thisbeating phenomenon reduced gradually and the closer theexciting frequency approached to the first natural frequencyof the cantilever system the faster the amplitude of thebeating decreased

Furthermore the amplitudes of the theoreticalwaveformsin Figures 11 and 12 are different from the experimental resultin Figure 13 This may because af the following (a) The effectof mechanical damping with respect to stiffness was onlyconsidered and the mass damping was ignored in theoreticalanalysis which can also weaken the transient vibration faster(b) The first natural frequency of the cantilever system wascalculated and measured by theoretical and experimentalmethods as 815Hz and 812Hz respectively The little dif-ference made that the max amplitudes from theoretical andexperimental results did not appear at the same frequency(c) In theoretical analysis the tip-mass was equated by theinertia force and inertia moment acting at the free end ofthe cantilever so the total length for theoretical analysiswas just the length of the beam of 1204mm Howeverduring experiments the tip-displacement was measured at

12 Shock and Vibration

005

0

minus0050 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6

0 1 2 3 4 5 6 0 1 2 3 4 5 6

0 1 2 3 4 5 6 0 1 2 3 4 5 6

0 1 2 3 4 5 6

0

0

1 2 3 4 5 6

02

01

0

minus01

minus02

02

01

0

0

minus01

minus02

015

01

005

minus005

minus01

008

006

004

002

0

minus002

minus004

minus006

minus008

008

006

004

002

0

minus002

minus004

minus006

minus008

008

006

004

002

minus002

minus004

minus006

minus008

2

15

1

05

0

minus05

minus1

minus15

minus2

01

005

0

minus005

minus01Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at735

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

795

Hz (

mm

)

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at855

Hz (

mm

)

tran

sient

to st

eady

at755

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mTh

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

815

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

875

Hz (

mm

)

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at775

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

835

Hz (

mm

)

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at895

Hz (

mm

)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Figure 11 Theoretical tip displacement responses with the second-type mode shape of the beam from transient to steady state

the approximate position of 119909 = 1204mm This is justan approximate value because Eddy-Current Sensor cannotgive exact measurement results of the displacement at thejunction between the cantilever and the tip-mass This maycause that the amplitude level from theoretical discussionwashigher than from experimental measurement In additionthe amplitude of the instantaneous base displacement fromthe exciter may be not the fixed value 001mm at the startinginstant Therefore the maximum displacement responsesduring the transient state in the experimental measurementsmay shift away from the real experimental results

The theoretical responses with two improved types ofmode shapes and the experimental displacement responsesversus varying positions of beam and the frequencies atsteady state are shown in Figures 14ndash16 respectively Figures14 and 15 show two types of theoretical displacements atthe positions of 119909 = 364mm 574mm 784mm 994mmand 1204mmwith continuous varying frequencies In exper-iments the displacement responses were measured at thesame positions with theoretical calculations but with somecertain frequencies of 782Hz 792Hz 802Hz 807Hz81Hz 812Hz 817Hz 82Hz 83Hz and 84Hz as plottedin Figure 16 It is noticed that when the frequency of baseexcitation approached to the first natural frequency of thecantilever system the amplitude of forced vibration willrapidly increase and become maximum for the case when

the frequency of the excitation coincides with the naturalfrequency Table 3 lists the values of the first natural frequen-cies and the displacement responses from theoretical andexperimental results at some certain positions long the beamunder the first natural frequencies It can be seen that thetheoretical and experimental results are matched very close

Table 3 The first natural frequencies and the resonantdisplacements at some certain positions

In a word the theoretical results with two different modeshapes derived by two improved analytical methods haveextremely the same behaviors at transient and steady stateswith respect to varying frequencies from base excitationThedisplacement responses indicate close agreement betweenthe experimental and analytical results All of these resultsgive full verification for the improvement of the analyticalmethods in this paper utilized in recalculating the modeshapes of a cantilever beam when an offset distance existsbetween the gravity center of the tip-mass and the point ofattachment

5 Conclusions

This paper presents two improved analytical methods ofcalculation for natural frequencies and mode shapes of auniform cantilever beam carrying a tip-mass under an input

Shock and Vibration 13

005

0

minus005Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at735

Hz (

mm

)

0 1 2 3 4 5 6

Time (s)

0 1 2 3 4 5 6

0 1 2 3 4 5 6

02

01

0

minus01

minus02

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at795

Hz (

mm

)

0

015

01

005

minus005

minus01

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at855

Hz (

mm

)

0 1 2 3 4 5 6

Time (s)

008

006

004

002

0

minus002

minus004

minus006

minus008tran

sient

to st

eady

at755

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

m

2

15

1

05

0

minus05

minus1

minus15

minus2Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at815

Hz (

mm

)

0 1 2 3 4 5 6

008

006

004

002

0

minus002

minus004

minus006

minus008Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at875

Hz (

mm

)

0 1 2 3 4 5 6

01

005

0

minus005

minus01Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at775

Hz (

mm

)

0 1 2 3 4 5 6

Time (s)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

0 1 2 3 4 5 6

0 1 2 3 4 5 6

0

008

006

004

002

minus002

minus004

minus006

minus008Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at895

Hz (

mm

)

02

01

0

minus01

minus02

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at835

Hz (

mm

)Figure 12 Theoretical tip displacement responses with third-type mode shape of the beam from transient to steady state

Table 3 The first natural frequencies and the resonant displacements at some certain positions

Resonant frequency or positions The values of frequencies and displacement responses (Hz or mm) ErrorsTheory based on 119880(2)

1(119909) Theory based on 119880(3)

1(119909) Experiment

The 1st natural frequency 815 815 812 037119909 = 364mm 01285 01285 01277 062119909 = 574mm 03022 03022 03001 069119909 = 784mm 05279 05279 05246 063119909 = 994mm 07905 07905 07855 063119909 = 1204mm 10742 10742 10671 066

base excitation using forced vibration theory and the methodof separation of variables A simplified cantilever model byreplacing the tip-mass with an equivalent inertial force andinertial moment acting at the free end of the cantilever beamwas detailed In addition using the Dirac Delta Functionthe base transverse excitation the concentrated equivalentinertia force and inertia moment were represented in theform of distributed loads which can make the method ofseparation of variables be utilized properly In this case sometypical results of natural frequencies and mode shapes of thecantilever beam with a tip-mass were derived using the rein-forced method of separation of variables and the developedforced vibration respectively Furthermore based on modalanalysis method the transient and steady displacement

responses of the cantilever system with a tip-mass and a sinebase excitation were presented in which two different types ofmode shapes calculated by two improved analytical methodswere used Finally an experimental cantilever system wasfabricated and some practical displacement responses weremeasured

From the results of the typical natural frequencies andmode shapes we can easily found that the mode shapesfrom the first improved type were identical to the modeshapes from the second type whether the mode shapes fromthe lower order or from the higher order even the offsetdistance was large The comparing results of the transientand steady displacement responses from the theoreticalanalysis and experimental measurements further proved the

14 Shock and Vibration

005

0

minus0050 2 4 6 8 10 12

0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12

0 2 4 6 8 10 120 2 4 6 8 10 12

0 2 4 6 8 10 12 0 2 4 6 8 10 120 2 4 6 8 10 12

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at732

Hz (

mm

)

0

008

006

004

002

minus002

minus004

minus006

minus008

0

008

006

004

002

minus002

minus004

minus006

minus008

0

008

006

004

002

minus002

minus004

minus006

minus008

0

015

01

005

minus005

minus01

01

005

0

minus005

minus01

2

15

1

05

0

minus05

minus1

minus15

minus2

03

02

01

0

minus01

minus02

03

02

01

0

minus01

minus02

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at752

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at772

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at792

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at812

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at832

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at852

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at872

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at892

Hz (

mm

)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Figure 13 Experimental displacement responses at the position of 119909 = 1204mm from transient to steady state

12

1

08

06

04

02

0120 100 80 60 40 20

78

80

82

84Theo

retic

al st

eady

disp

lace

men

t (m

m)

Position at the beam (mm)Freq

uency

(Hz)

Figure 14 Theoretical steady-displacement responses versus posi-tions and frequencies with mode 119880(2)

1(119909)

improvements of the analytical methods for calculating themode shapes of the cantilever beam with a tip-mass anda base excitation In addition the amplitudes of the modeshapes decreased with the increasing mass value of the tip-mass when the moment of inertia and offset were constantsThe amplitude of the first-order mode shape increased but

12

1

08

06

04

02

0120 100 80 60 40 20

78

80

82

84Theo

retic

al st

eady

disp

lace

men

t (m

m)

Position at the beam (mm)Freq

uency

(Hz)

Figure 15 Theoretical steady-displacement responses versus posi-tions and frequencies with mode 119880(3)

1(119909)

the other nine mode shapes decreased with the increasingmoment of inertia when the mass and offset were set asconstants Conversely the amplitude of the first mode shapedecreased but the other ninemode shapes increased with theincreasing offset distance under the condition that the massand moment of inertia were constants Globally the effect

Shock and Vibration 15

12

1

08

06

04

02

0

Expe

rimen

tal s

tead

y di

spla

cem

ent (

mm

)

120 100 80 60 40 20Position at the beam (mm)

78

80

82

84Freq

uency

(Hz)

Figure 16 Experimental steady-displacement responses versuspositions and frequencies

from varying combinations of the mass and moment inertiaof the tip-mass and the offset distance for the lower-ordermode shapes was stronger than for the higher ones

Furthermore it is clearly noticed that the mass of tip-mass can reduce the values of the natural frequencies andthe increasing mass value gave the decreasing frequencieswhen the moment of inertia of the tip-mass and the offsetdistance were set as constants Similarly the moment ofinertia can further decrease the natural frequencies and thehigher moment of inertia resulted in the lower frequencieswhen the mass and the offset were constants Differently theoffset distance can reduce the first-order natural frequenciesbut increase the other nine order frequencies and the higheroffset distance led to the lower first-order natural frequenciesbut produced the higher other nine order frequencies whenthe mass and the moment of inertia were set to constant Inglobal view the influence from the varying mass momentof inertia and offset distance for the lower-order naturalfrequencies was larger than for the higher-order ones weatherthis influence was negative or positive

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the Natural Science Foundationsof China (no 51275380) and the Scientific and TechnologicalProject of Shaanxi Province (no 2012K06-36) The authorswould like to thank the reviewers for their valuable commentson this paper

References

[1] P A A Laura J L Pombo and E A Susemihl ldquoA note on thevibrations of a clamped-free beam with a mass at the free endrdquoJournal of Sound and Vibration vol 37 no 2 pp 161ndash168 1974

[2] B Rama Bhat andHWagner ldquoNatural frequencies of a uniformcantilever with a tip mass slender in the axial directionrdquo Journalof Sound and Vibration vol 45 no 2 pp 304ndash307 1976

[3] C W S To ldquoVibration of a cantilever beam with a baseexcitation and tip massrdquo Journal of Sound and Vibration vol83 no 4 pp 445ndash460 1982

[4] P A A Laura ldquoComments on lsquoVibration of a cantilever beamwith a base excitation and tip massrsquordquo Journal of Sound andVibration vol 88 no 4 p 569 1983

[5] C W S To ldquoDiscussion on lsquovibration of a cantilever beam witha base excitation and tip massrsquordquo Journal of Sound and Vibrationvol 97 no 2 pp 349ndash351 1984

[6] R G Jacquot ldquoFurther comments on lsquoVibration of a cantileverbeam with a base excitation and tip massrsquordquo Journal of Sound andVibration vol 93 no 2 pp 312ndash313 1984

[7] S AMousavi Lajimi andG R Heppler ldquoComments on lsquonaturalfrequencies of a uniform cantilever with a tip mass slender inthe axial directionrsquordquo Journal of Sound and Vibration vol 331 no12 pp 2964ndash2968 2012

[8] R B Bhat ldquoAuthors response to the comments on lsquonaturalfrequencies of a uniform cantilever with a tip mass slender inthe axial directionrsquordquo Journal of Sound and Vibration vol 331 no12 pp 2962ndash2963 2012

[9] E Esmailzadeh and G Nakhaie-Jazar ldquoPeriodic behavior ofa cantilever beam with end mass subjected to harmonic baseexcitationrdquo International Journal of Non-Linear Mechanics vol33 no 4 pp 567ndash577 1998

[10] H Abramovich and O Hamburger ldquoVibration of a cantilevertimoshenko beam with a tip massrdquo Journal of Sound andVibration vol 148 no 1 pp 162ndash170 1991

[11] B Pratiher ldquoVibration control of a transversely excited can-tilever beam with tip massrdquo Archive of Applied Mechanics vol82 no 1 pp 31ndash42 2012

[12] M Kim M Hoegen J Dugundji and B L Wardle ldquoModelingand experimental verification of proof mass effects on vibrationenergy harvester performancerdquo Smart Materials and Structuresvol 19 Article ID 045023 2010

[13] Y T Li and R Wong ldquoIntegral and series representations ofthe Dirac delta functionrdquo Communications on Pure and AppliedAnalysis vol 7 no 2 pp 229ndash247 2008

[14] M Al-Gwaiz Sturm-LiouvilleTheory and its Applications chap-ter 3 Springer London UK 2008

[15] F Bao-lian ldquoGeneralized reciprocal theorems and their appli-cationsrdquo Applied Mathematics and Mechanics vol 23 no 2 pp203ndash210 2001

[16] C Cai H Zheng M Khan and K HungModeling of MaterialDamping Properties in ANSYS Defense Systems Division Insti-tute of High Performance Computing 2002

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International Journal of

Shock and Vibration 5

Under the simple harmonic excitation 119890119895120596119905 the steady statesolution of (23) can be regarded with the form as

119906119886(119909 119905) = 119880

119886(119909 120596) 119890

119895120596119905minus120593

= [1198611cos (120573119909) + 119861

2sin (120573119909)] 119890119895120596119905minus120593

+1198613cosh (120573119909) + 119861

4sinh (120573119909)]

(28)

where 119861119899(119899 = 1 2 3 4) are arbitrary constants the

parameter 120573 is defined by 1205734 = 1198981205962(119864119868) 119895 is the imaginaryunit and 120593 is the lagging phase

We recalculated the natural frequencies and the modeshapes directly by substituting (28) into four boundaryconditions (24) through (27) and obtained the characteristicequation and displacement response which are different from[3]

119863 = 1 + 119888119888ℎ minus 120601119906 (119904119888ℎ minus 119888119904ℎ)

minus 1198771[119904119888ℎ + 119888119904ℎ minus 120601119906 (1 minus 119888119888ℎ)]

minus 21198772119904119904ℎ minus 119877

2

2(1 minus 119888119888ℎ)

119880 (119909) = (

119861

2119863

) 1198651015840

1119906cos (120573119909) + 1198651015840

2119906cosh (120573119909)

+1198651015840

3119906[sin (120573119909) minus sinh (120573119909)]

(29)

where 119861 is defined by 119887(119905) = 119861119890119895120596119905 and in which

1198651015840

1119906= 1+119888119888ℎminus119904119904ℎminus2120593119906119904119888ℎminus2119877

1119888119904ℎ+(119877

1120593119906minus119877

2

2)(1minus

119888119888ℎ + 119904119904ℎ) minus 21198772(119888119888ℎ + 119904119904ℎ) =119865

1119906minus 21198772(119888119888ℎ + 119904119904ℎ)

1198651015840

2119906= 1+119888119888ℎ+119904119904ℎ+2120593119906119888119904ℎminus2119877

1119904119888ℎ+(119877

1120593119906minus119877

2

2)(1minus

119888119888ℎ minus 119904119904ℎ) + 21198772(119888119888ℎ minus 119904119904ℎ) =119865

2119906+ 21198772(119888119888ℎ minus 119904119904ℎ)

1198651015840

3119906= (119904119888ℎ + 119888119904ℎ)(1 minus 119877

1120593119906 +119877

2

2) + 2[120593119906119888119888ℎ minus 119877

1119904119904ℎ minus

1198772(119904119888ℎ minus 119888119904ℎ)] = 119865

3119906

It is obvious that the characteristic equation derived by thereinforced method of separation of variables is equal tothe equation obtained by the developed method of forcedvibration

Themode shapes cannot be obtained directly because theboundary conditions contain the parameter1205962 However thisproblem can be successfully solved by Dirac Delta Function[13] Thus the mode shapes may be obtained by the Sturm-Liouville theory [14] and expressed as

119880119894(119909) = (

1

2

) 1198651015840

1119906119894119865 cos (120573

119894119909) + 119865

1015840

2119906119894cosh (120573

119894119909)

+ 1198651015840

3119906119894[sin (120573

119894119909) minus sinh (120573

119894119909)]

(30)

3 Computed Results and Comparisons

In this section three types of the natural mode shapes ofthe cantilever beam with a tip-mass were computed and

compared The first one is called the undeveloped modeshapes which is directly obtained from [3] by the expression

119880(1)

119894(119909) = (

1

2

) 1198651119894119888

cos(119906119894119909

119871

) + 1198652119894119888

cosh (119906119894119909

119871

)

+ 1198653119894119888[sin(

119906119894119909

119871

) minus sinh(119906119894119909

119871

)]

(31)

where 1198651119894119888

= 1198651119894119888ℎ(119906119894) 1198652119894119888

= 1198652119894119888ℎ(119906119894) and 119865

3119894119888=

1198653119894119888ℎ(119906119894) in which 119865

1119894 1198652119894 and 119865

3119894are the functions 119865

1119906 1198652119906

and 1198653119906

with the variable 119906 replaced by 119906119894 respectively

The second one is based on the expression of thedeveloped mode shapes derived in this paper Let (30) bemultiplied by 21198651015840

3119894to give the second mode shapes as

119880(2)

119894(119909) = sin(

119906119894119909

119871

) minus sinh(119906119894119909

119871

) + (

1198651015840

1119894

1198651015840

3119894

) cos(119906119894119909

119871

)

+ (

1198651015840

2119894

1198651015840

3119894

) cosh (119906119894119909

119871

)

(32)

At last the arbitrary constant 1198622in (22) was chosen as

constant 1 to give the third type of the natural mode shapes as

119880(3)

119894(119909) = sin(

119906119894119909

119871

) minus 119904ℎ (

119906119894119909

119871

)

minus 120582 [cos(119906119894119909

119871

) minus 119888ℎ (

119906119894119909

119871

)]

(33)

Before the mode shapes are determined the roots ofthe characteristic equation must be calculated firstly Thecharacteristic equation (21) or 119863 = 0 can be solved by twosteps First approximate roots were pointed out by applyinggraphing method Then the approximate roots were iteratedby the bisection method to give the exact characteristic rootswith given error All these processes were finished usingMATLAB software

Because the condition that the gravity center of the tip-mass coincided with the point of the end attachment hasbeen verified in [6] we focus on the condition when theoffset distance is not zero Table 1 shows the first five roots ofthe characteristic equations from five samples with differentvalues of 120601 119896119871 and 119889119871 It can be clearly noticed thatthe larger value of 119889119871 resulted in the less value of the firstcharacteristic root but the larger values of the other fourcharacteristic roots when 120601 and 119896119871 were set as constants

Once the roots of the characteristic equation have beenobtained three types of the first five mode shapes of thecantilever structure from the five given samples can becalculated Figures 4 5 6 7 and 8 show the first five modeshapes of the cantilever with various combinations of 120601 119896119871and 119889119871

6 Shock and Vibration

Table 1 Typical roots of characteristic equation with variables 120601 119896119871 and 119889119871

Sample 120601 119870119871 119889119871

Roots of characteristic equation1199061

1199062

1199063

1199064

1199065

1 4 4 0 035211 129234 477818 788401 11017802 4 4 05 034814 130637 477952 788472 11018273 4 4 1 034200 132825 478225 788634 11019404 4 4 2 032544 138983 479177 789233 11023685 4 4 4 028806 154543 482543 791455 1103990

003002001

0

0 05 1 0 05 1 0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

000000

050000

100000

i = 1

000000

050000

100000

i = 1

000000 000000

000000

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

000000

050000

100000

i = 2

000000

050000

100000

i = 2

minus186571 times 10minus9

000792

003152

minus198560 times 10minus8

minus9418658

minus18545378

0

minus100

minus2000 05 1

0 05 1

0 05 1

0 05 1

1050

times104 i = 3

000000

050000

100000

i = 3

000000

050000

100000

i = 3

786191 times 10minus6

111026 times 104

minus686262 times 103

50

minus5

times105 i = 4000000

050000

100000

i = 4

000000

050000

100000

i = 4

i = 5000000

050000

100000

i = 5

000000 000000

050000

100000

i = 5times106

20

minus2

000017

minus271300 times 106

minus855944 times 104

1

05

0

1

05

0

03

02

01

0

03

02

01

0

151

050

151

050

2

0

minus2

2

0

minus2

210

minus1

210

minus1

minus493414 times 10minus8

032943

131050

367824 times 10minus11

017448

034354

113682 times 10minus10

160541

minus009923

457130 times 10minus11

minus002114

006144

866744 times 10minus11

minus140614

minus004436

032943

131050

017448

034354

160541

minus009923

000000

minus002114

006144

minus140614

minus004436

Notes at xL Notes at xL Notes at xLValues ValuesValues

Ui(x)

xL

Ui(x)

xL

Ui(x)

xL

minus233509 times 10minus5

107963 times 104

minus313859 times 104

First-type mode shapes U(1)i

(x) Second-type mode shapes U(2)i

(x) Third-type mode shapes U(3)i

(x)

Figure 4 Three types of the first five mode shapes with the combinations of 120601 = 4 119896119871 = 4 and 119889119871 = 0

It is obvious that when 119889119871 = 0 the first type of naturemode shapes 119880(1)

119894(119909) derived by the undeveloped method

turned far away from the third-type mode shapes 119880(3)119894(119909)

derived by the reinforced method of separation of variablesIn addition the larger value of 119889119871 resulted in the moreinconsistent mode shapes However the second type of modeshapes 119880(2)

119894(119909) obtained by the developed method of forced

vibration seemed in good agreement with the third onesFurthermore the amplitude of the first mode shape

decreased but the other four mode shapes increased with theincreasing value of the 119889119871 when 119896119871 and 120601 were fixed to be

constants The effect from the varying value of 119889119871 for thelower mode shapes was always stronger than for the higher

4 Experimental Verification

In order to experimentally explain and compare two typesof natural mode shapes derived by two improved methodsabove the transient and steady responses of the cantileversystem under an input base excitation should be calculatedfirst This whole calculation can be finished by method ofmodal analysis But above all the orthogonality property of

Shock and Vibration 7

0 05 1

0 05 1

Ui(x)

xL

Ui(x)

xL

Ui(x)

xL

014013012011

0

minus100

minus200

000000

050000

100000

i = 1

000000

050000

100000

i = 2

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

011044

011840

014143

533123

minus9287788

minus18666064

0 05 1

1050

times104

000000

050000

100000

i = 3minus8449356

107519 times 105

minus970528 times 103

0 05 1

50

minus5

times105

000000

050000

100000

i = 422493344

139452 times 104

139108 times 105

0 05 1

40

minus4minus8

times106

000000

050000

100000

i = 5minus42806116

minus266045 times 106

minus765307 times 106

0 05 1

000000

050000

100000

i = 106

04

02

0

392120 times 10minus9

017490

069258

0 05 1

06

04

02

0

0 05 1

000000

050000

100000

i = 2547973 times 10minus11

017881

034942

03

02

01

00 05 1

03

02

01

0

0 05 1

000000

050000

100000

i = 3minus505260 times 10minus11

160517

minus010133

151

050

0 05 1

151

050

0 05 1

000000

050000

100000

i = 4minus776048 times 10minus11

minus002162

006261

2

0

minus20 05 1

2

0

minus2

0 05 1

000000

050000

100000

i = 5663249 times 10minus11

minus140614

minus004516

210

minus1

000000

000000

000000

000000

017490

069258

017881

034942

160517

minus010133

minus002162

006261

000000

050000

100000

i = 5

0 05 1

210

minus1

000000

minus140614

minus004516

Notes at xL Notes at xL Notes at xLValues ValuesValuesFirst-type mode shapes U(1)i

(x) Second-type mode shapes U(2)i

(x) Third-type mode shapes U(3)i

(x)

Figure 5 Three types of the first five mode shapes with the combinations of 120601 = 4 119896119871 = 4 and 119889119871 = 05

034

033

032

031

0 05 1 0 05 1 0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1 0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0

minus100

minus200

10

5

0

times104

50

minus5

times105

times106

0minus5minus10minus15

Ui(x)

xL

Ui(x)

xL

Ui(x)

xL

04

02

0

04

02

0

04

02

0

04

02

0

151

050

151

050

2

0

minus2

2

0

minus2

210

minus1

210

minus1

000000

050000

100000

i = 1

000000 000000

050000

100000

i = 1

030899

031712

033934

000000

050000

100000

i = 2861965

minus9105604

minus18440224

000000

050000

100000

i = 3minus10637851

973291 times 104

minus123754 times 104

000000

050000

100000

i = 419388076

168672 times 104

311790 times 105

000000

050000

100000

i = 5minus19120922

minus244086 times 106

minus152554 times 107

000000

050000

100000

i = 1

331605 times 10minus9

012125

047836

000000

050000

100000

i = 2

minus550109 times 10minus11

018658

036219

000000

050000

100000

i = 3

minus175471 times 10minus11

160459

minus010631

000000

050000

100000

i = 4

109777 times 10minus10

minus002275

006562

000000

050000

100000

i = 5

836395 times 10minus11

minus140614

minus004731

012125

047836

000000 000000

050000

100000

i = 2

018658

036219

000000 000000

050000

100000

i = 4

minus002275

006562

000000 000000

050000

100000

i = 5

minus140614

minus004731

000000 000000

050000

100000

i = 3

160459

minus010631

Notes at xL Notes at xL Notes at xLValues ValuesValuesFirst-type mode shapes U(1)i

(x) Second-type mode shapes U(2)i

(x) Third-type mode shapes U(3)i

(x)

Figure 6 Three types of the first five mode shapes with the combinations of 120601 = 4 119896119871 = 4 and 119889119871 = 1

8 Shock and Vibration

088087086085

0 05 1 0 05 1 0 05 1

0 05 1 0 05 1

0 05 10 05 1

0 05 1 0 05 1

0 05 1 0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0

minus50

minus100

minus150

6420

minus2

times104

840

minus4

times105

0minus1minus2minus3

times107

0302010

03

02

01

0

04

02

0

04

02

0

151

050

151

050

210

minus1

210

minus1

210

minus1

210

minus1

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

084906

085736

087731

minus1308377

minus9308045

minus16072843

63490847

565642 times 104

minus172997 times 104

minus329762092

230021 times 104

667852 times 105

946242416

minus149912 times 106

minus306922 times 107

minus995048 times 10minus10

007811

030669

511030 times 10minus11

021118

040600

minus114494 times 10minus10

160244

minus012466

688252 times 10minus11

minus002686

007708

183045 times 10minus11

minus140613

minus005561

000000

000000

000000

000000

000000

007811

030669

021118

040600

160244

minus012466

minus002686

007708

minus140613

minus005561

Ui(x)

xL

Ui(x)

xL

Ui(x)

xL

Notes at xL Notes at xL Notes at xLValues ValuesValuesFirst-type mode shapes U(1)i

(x) Second-type mode shapes U(2)i

(x) Third-type mode shapes U(3)i

(x)

Figure 7 Three types of the first five mode shapes with the combinations of 120601 = 4 119896119871 = 4 and 119889119871 = 2

the natural mode shapes should be derived In fact usingBetti theorem [15] the orthogonality properties with respectto mass and stiffness can be obtained respectively as follows

int

119871

0

119898119880(119896)

119903(119909)119880(119896)

119894(119909) 119889119909 +119872119880

(119896)

119903(119871)119880(119896)

119894(119871)

+ 119903119888119872119880(119896)

119894(119871)

119889119880(119896)

119903(119871)

119889119909

+ 119903119888119872119880(119896)

119903(119871)

119889119880(119896)

119894(119871)

119889119909

+ (119869 + 1199032

119888119872)

119889119880(119896)

119894(119871)

119889119909

119889119880(119896)

119903(119871)

119889119909

= (120575(119896)

119903119894)

2

(119896 = 1 2 119903 119894 = 1 2 )

(34)

int

119871

0

119864119868

1198892119880(119896)

119894(119909)

1198891199092

1198892119880(119896)

119903(119909)

1198891199092

119889119909 = (120596119903120575(119896)

119903119894)

2

(119896 = 1 2 119903 119894 = 1 2 )

(35)

In this case the normalized mode shapes can be defined as

119880(119896)

119903119873(119909) =

119880(119896)

119903(119909)

120575(119896)

119903119903

(36)

where

(120575(119896)

119903119903)

2

= int

119871

0

119898[119880(119896)

119903(119909)]

2

119889119909 +119872[119880(119896)

119903(119871)]

2

+ 2119903119888119872119880(119896)

119903(119871) [

119889

119889119909

119880(119896)

119903(119871)]

+ (119869 +1198721199032

119888) [

119889

119889119909

119880(119896)

119903(119871)]

2

(37)

41 Theoretical Responses of the Cantilever System with a BaseSine Excitation According to the method of modal analysisthe solution of (8) has the form of

119906119886(119909 119905) =

infin

sum

119903=1

119880(119896)

119903119873(119909) 119902119903(119905) (38)

So inserting (38) into (8) gives

119864119868

infin

sum

119903=1

1198894119880(119896)

119903119873(119909)

1198891199094

119902119903(119905) + 119898

infin

sum

119903=1

119880(119896)

119903119873(119909)

1198892119902119903(119905)

1198891199052

= [119872120575 (119909 minus 119871) + 119903119888119872

119889120575 (119909 minus 119871)

119889119909

minus 119898]

1198892119887 (119905)

1198891199052

(39)

Considering the orthogonality properties of equations (34)and (35) let (39) be multiplied by119880(119896)

119894119873(119909) and integrated over

Shock and Vibration 9

192

191

19

0 05 1 0 05 1

0 05 1 0 05 1

0 05 1

0 05 10 05 1

0 05 10 05 1

0 05 1 0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0

minus200

minus400

minus600

0minus5minus10

times104

1

0

minus1

times106

times107

0minus2minus4minus6

02

01

0

02

01

0

06

04

02

0

06

04

02

0

15

1

05

0

15

1

05

0

210

minus1

210

minus1

210

minus1

210

minus1

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

000000

050000

100000

i = 1

000000

050000

100000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

189913

190621

192051

minus56420429

minus28138349

4723398

134513 times 104

minus110859 times 105

minus260952 times 104

minus615688 times 104

496921 times 104

145091 times 106

168417 times 105

241806 times 106

minus633899 times 107

137503 times 10minus9

004964

019458

minus238241 times 10minus11

028530

054402

159443

minus019150

minus283947 times 10minus11

minus004212

012050

771494 times 10minus11

minus140605

minus008755

000000

000000

000000 000000

000000

000000

004964

019458

028530

054402

159443

minus019150

minus004212

012050

minus140605

minus008755

Ui(x)

xL

Ui(x)

xL

Ui(x)

xL

Notes at xL Notes at xL Notes at xLValues ValuesValues

minus387890 times 10minus11

First-type mode shapes U(1)i

(x) Second-type mode shapes U(2)i

(x) Third-type mode shapes U(3)i

(x)

Figure 8 Three types of the first five mode shapes with the combinations of 120601 = 4 119896119871 = 4 and 119889119871 = 4

the domain 0 lt 119909 lt 119871 to give a set of independent ordinarydifferential equations as

1198892119902119894(119905)

1198891199052

+ 1205962

119894119902119894(119905) = 119891

(119896)

119894(119905) (40)

Considering the properties of Dirac Delta Function

int

119871

0

119880(119896)

119894119873(119909) 120575 (119909 minus 119871) 119889119909 = 119880

(119896)

119894119873(119871)

int

119871

0

119880(119896)

119894119873(119909)

119889120575 (119909 minus 119871)

119889119909

119889119909 = minus

119889119880(119896)

119894119873(119871)

119889119909

(41)

the generalized forces associated with the generalized coordi-nates 119902

119894(119905) can be obtained as

119891(119896)

119894(119905)

= [119872119880(119896)

119894119873(119871) minus 119903

119888119872

119889119880(119896)

119894119873(119871)

119889119909

minus 119898int

119871

0

119880(119896)

119894119873(119909) 119889119909]

1198892119887 (119905)

1198891199052

(42)

Furthermore considering the mechanical damping asRayleigh damping and ignoring the part of mass dampingthe damping coefficient 120577 of the bimorph beam can be deter-mined by the modal damping ratio which can be obtained

by experiment [16] The independent ordinary differentialequations with damping then can be written as

1198892119902119894(119905)

1198891199052

+ 2120596119894120577

119889119902119894(119905)

119889119905

+ 1205962

119894119902119894(119905)

= [119872119880(119896)

119894119873(119871) minus 119903

119888119872

119889119880(119896)

119894119873(119871)

119889119909

minus 119898int

119871

0

119880(119896)

119894119873(119909) 119889119909]

1198892119887 (119905)

1198891199052

(43)

Now (43) can be solved using Laplace transform Applyinga sine excitation 119887(119905) = 119861 sin(120596119905) to the base of cantileverstructure under zero initial conditions the transfer functionof the independent differential equations for the cantileverbeam with tip-mass under base displacement excitation canbe written as

119866119894(119904)

=

119876119894(119904)

119861119894(119904)

=

(minus1205962)[119872119880

(119896)

119894119873(119871)minus119903

119888119872(119889119880

(119896)

119894119873(119871) 119889119909)minus119898int

119871

0119880(119896)

119894119873(119909) 119889119909]

1199042+ 2120596119894120577119904 + 120596

2

119894

(44)

10 Shock and Vibration

where 119876119894(119904) and 119861

119894(119904) are the Laplace transforms of 119902

119894(119905) and

119887(119905) respectively The inverse transformation of the transferfunction will be further obtained as

119892119894(119905)

=

(minus1205962)[119872119880

(119896)

119894119873(119871)minus119903

119888119872(119889119880

(119896)

119894119873(119871) 119889119909)minus119898int

119871

0119880(119896)

119894119873(119909) 119889119909]

120596119894radic1 minus 120577

2

times 119890minus120596119894120577119905 sin(120596

119894radic1 minus 120577

2119905)

(45)

In this case the solutions of (43) can be solved by convolutiontheorem as

119902119894(119905) = 119887 (119905) lowast 119892

119894(119905) = int

119905

0

119887 (120591) 119892119894(119905 minus 120591) 119889120591 (46)

Finally substituting the result of (46) into (38) and after somelengthy algebraic manipulation the absolute displacementresponse of the cantilever beam with tip-mass under basetransverse excitation can be obtained as

119906119886(119909 120596 119905) =

infin

sum

119894=1

119880(119896)

119894119873(119909)

119861 [119872119880(119896)

119894119873(119871) minus 119903

119888119872(119889119880

(119896)

119894119873(119871) 119889119909) minus 119898int

119871

0119880(119896)

119894119873(119909) 119889119909]

radic1205964+ 2 (2120577

2minus 1) 120596

21205962

119894+ 1205964

119894

times [

1205963

120596119894radic1 minus 120577

2

119890minus120596119894120577119905 sin(120596

119894radic1 minus 120577

2119905 + 1206011) + 1205962 sin (120596119905 + 120601

2)]

(47)

where 1206011and 120601

2are determined by tan(120601

1) = 2120577radic1 minus 120577

2

((120596120596119894)2+ 21205772minus 1) and tan(120601

2) = minus2120577(120596120596

119894)(1 minus (120596120596

119894)2)

It can be seen that the solution consists of two parts abouttimeThe first term represents free vibration which will decaywith time exponentially The second term depending on theexcitation represents the forced vibration of the system

42 Experimental Responses and Comparisons with the Theo-retical Results In order to explain the absolute displacementresponse of the cantilever structure under base excitationexperimentally a typical cantilever beam in which theelongated beam was inserted and fixed into the tip-mass tomake sure the gravity center of tip-mass was collinear withthe beam was fabricated (as shown in Figure 9) The chosengeometric and physical properties of the cantilever beamincluding the tip-mass are given in Table 2

The complete experimental devices are shown inFigure 10 The electromagnetic vibrating table type SW-TFAcontrolled by a programmable controller type SW-F-2TF-2was used to create a specific harmonic input excitationThe cantilever beam with a tip-mass was clamped onthe vibrating top platform Eddy-Current Sensors typeDM6008 and ZA210800 were used to measure the inputdisplacement from base vibrating exciter and the absolutedynamic displacement from the beam respectively Allmeasurements from cantilever beam and vibrating exciterwere connected to Digital Oscilloscope type DS1204Band the Industrial Computer including a PC-based DataAcquisitionModule from ADLINK company type DAQ2213The Digital Oscilloscope and Industrial Computer displayedand processed the measurement results using the DAQPilotsoftware and the results from subsequent analysis using theMATLAB software

Table 2 Characteristic properties of the cantilever beam includingtip-mass

Physical properties Beam Tip-massYoungrsquos modulus 119864 (GPa) 69 mdashDensity 120588

119904or 120588119898(kgm3) 2630 7753

Length 119871 or 119871119898(mm) 1204 156

Thickness 119905119904or 119905119898(mm) 49 295

Width 119908 (mm) 102 102Total mass119872 (g)a mdash 3383Offset 119903

119888(mm)b mdash 801

Rotary inertia 119869 (kgsdotm2)b c mdash 333 times 10minus6aThe total mass of the tip-mass including the elongated part of beam wascalculated by119872 = 119871

119898119905119898119908120588119898+ 119871119904119905119904119908(120588119904minus 120588119898) where 119871

119904= 10mm

bThe offset distance and the rotary inertia were calculated according to thegeometry and material properties of the beam and tip-mass using equations119903119888= (120588

1198981199051198981198712

119898minus 1205881198981199051199041198712

119904+ 1205881199041199051199041198712

119904)(2[120588

119898119905119898119871119898+ 119871119904119905119904(120588119904minus 120588119898)])

119869 = 120588119898119871119898119905119898119908 [(112)(119871

2

119898+ 1199052

119898) + ((12)119871

119898minus 119903119888)2] + 119871

119904119905119904119908(120588119904minus

120588119898)[(112)(119871

2

119904+ 1199052

119904) + (119903119888minus (12)119871

119904)2]

cThe corresponding parameters 120601 119896119871 and 119903119888119871 were calculated as follows

120601 = 119872(120588119904119905119904119908) = 21377 119896119871 = radic119869119872119871 = 00824 and 119903

119888119871 = 00665

The analytical model defined by (47) in which two typesof mode shapes derived by two improved analytical methodsare utilized can be used to analyze the steady and transientdisplacement responses of the cantilever beam with a tip-mass and an input base excitation based on the chosenvarying frequencies time and positions along the lengthof the beam Typically the displacement responses frompositions along 119909-axis at 119909

1= 364mm 119909

2= 574mm

1199093

= 784mm 1199094

= 994mm and 1199095

= 1204mmoff the origin point (as shown in Figure 9) were measured

Shock and Vibration 11

y

b(t)

x1

x2

x3

x4

x5

Ls

L

t s

t m x

Lm

rc

Cg

o998400

Figure 9 Typical cantilever beam with a tip-mass

1 2

3

56

7

8

4

Tip-massCantilever beam

(7) Cantilever with a tip-mass

(1) Industrial computer including data acquisitionmodule type DAQ2213

(2) Digital oscilloscope type DS1204B(3) Programmable controller type SW-F-FTF-2(4) Power supply type DP1308A(5) Driver modules for Eddy-current sensors(6) Electromagnetic vibrating table type SW-TFA

(8) Eddy-current sensors type DM6008 and ZA210800

Figure 10 Complete experimental setup

by Eddy-Current Sensors The amplitude of the input sine-displacement excitation for analysis was measured as 119861 =

001mm and the damping ratio for the transverse vibrationaround the fundamental resonant frequency was found to be120577 = 00025 by Logarithmic Decrement Method

In the following paragraphs we focus on the transientand steady responses of the cantilever system with a sine baseexcitation

Figures 11 and 12 show the theoretical displacementresponses at the free end of the beam from transient tosteady state with two improved mode shapes above at thefirst resonant frequency It can be seen obviously that thedisplacement responses with two improved mode shapeswe proposed are almost identical This richly proves thatthe first-type mode shapes reformulated by the reinforcedmethod of separation of variables are equal to the second-type mode shapes derived by the developed method offorced vibration In addition it can be further obtainedthat when the frequency from the base excitation is veryclose but not exactly equal to the resonant frequency of freevibrations of the cantilever system a kind of vibration calledbeating occurs and the period of the beating increases as120596 approached 120596

1 This kind of beating decays exponentially

along the time due to the mechanical dampingFigure 13 shows the experimental displacement responses

at the position of 119909 = 1204mm from transient to steady

state As the same to the theoretical results experimental dis-placement response can be regarded as the sum of waveformsfrom transient and steady states But the transient responsevanished gradually with the time due to the mechanicaldamping and only the steady response left In addition thephenomenon of beating still existed The amplitude of thisbeating phenomenon reduced gradually and the closer theexciting frequency approached to the first natural frequencyof the cantilever system the faster the amplitude of thebeating decreased

Furthermore the amplitudes of the theoreticalwaveformsin Figures 11 and 12 are different from the experimental resultin Figure 13 This may because af the following (a) The effectof mechanical damping with respect to stiffness was onlyconsidered and the mass damping was ignored in theoreticalanalysis which can also weaken the transient vibration faster(b) The first natural frequency of the cantilever system wascalculated and measured by theoretical and experimentalmethods as 815Hz and 812Hz respectively The little dif-ference made that the max amplitudes from theoretical andexperimental results did not appear at the same frequency(c) In theoretical analysis the tip-mass was equated by theinertia force and inertia moment acting at the free end ofthe cantilever so the total length for theoretical analysiswas just the length of the beam of 1204mm Howeverduring experiments the tip-displacement was measured at

12 Shock and Vibration

005

0

minus0050 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6

0 1 2 3 4 5 6 0 1 2 3 4 5 6

0 1 2 3 4 5 6 0 1 2 3 4 5 6

0 1 2 3 4 5 6

0

0

1 2 3 4 5 6

02

01

0

minus01

minus02

02

01

0

0

minus01

minus02

015

01

005

minus005

minus01

008

006

004

002

0

minus002

minus004

minus006

minus008

008

006

004

002

0

minus002

minus004

minus006

minus008

008

006

004

002

minus002

minus004

minus006

minus008

2

15

1

05

0

minus05

minus1

minus15

minus2

01

005

0

minus005

minus01Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at735

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

795

Hz (

mm

)

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at855

Hz (

mm

)

tran

sient

to st

eady

at755

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mTh

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

815

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

875

Hz (

mm

)

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at775

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

835

Hz (

mm

)

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at895

Hz (

mm

)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Figure 11 Theoretical tip displacement responses with the second-type mode shape of the beam from transient to steady state

the approximate position of 119909 = 1204mm This is justan approximate value because Eddy-Current Sensor cannotgive exact measurement results of the displacement at thejunction between the cantilever and the tip-mass This maycause that the amplitude level from theoretical discussionwashigher than from experimental measurement In additionthe amplitude of the instantaneous base displacement fromthe exciter may be not the fixed value 001mm at the startinginstant Therefore the maximum displacement responsesduring the transient state in the experimental measurementsmay shift away from the real experimental results

The theoretical responses with two improved types ofmode shapes and the experimental displacement responsesversus varying positions of beam and the frequencies atsteady state are shown in Figures 14ndash16 respectively Figures14 and 15 show two types of theoretical displacements atthe positions of 119909 = 364mm 574mm 784mm 994mmand 1204mmwith continuous varying frequencies In exper-iments the displacement responses were measured at thesame positions with theoretical calculations but with somecertain frequencies of 782Hz 792Hz 802Hz 807Hz81Hz 812Hz 817Hz 82Hz 83Hz and 84Hz as plottedin Figure 16 It is noticed that when the frequency of baseexcitation approached to the first natural frequency of thecantilever system the amplitude of forced vibration willrapidly increase and become maximum for the case when

the frequency of the excitation coincides with the naturalfrequency Table 3 lists the values of the first natural frequen-cies and the displacement responses from theoretical andexperimental results at some certain positions long the beamunder the first natural frequencies It can be seen that thetheoretical and experimental results are matched very close

Table 3 The first natural frequencies and the resonantdisplacements at some certain positions

In a word the theoretical results with two different modeshapes derived by two improved analytical methods haveextremely the same behaviors at transient and steady stateswith respect to varying frequencies from base excitationThedisplacement responses indicate close agreement betweenthe experimental and analytical results All of these resultsgive full verification for the improvement of the analyticalmethods in this paper utilized in recalculating the modeshapes of a cantilever beam when an offset distance existsbetween the gravity center of the tip-mass and the point ofattachment

5 Conclusions

This paper presents two improved analytical methods ofcalculation for natural frequencies and mode shapes of auniform cantilever beam carrying a tip-mass under an input

Shock and Vibration 13

005

0

minus005Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at735

Hz (

mm

)

0 1 2 3 4 5 6

Time (s)

0 1 2 3 4 5 6

0 1 2 3 4 5 6

02

01

0

minus01

minus02

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at795

Hz (

mm

)

0

015

01

005

minus005

minus01

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at855

Hz (

mm

)

0 1 2 3 4 5 6

Time (s)

008

006

004

002

0

minus002

minus004

minus006

minus008tran

sient

to st

eady

at755

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

m

2

15

1

05

0

minus05

minus1

minus15

minus2Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at815

Hz (

mm

)

0 1 2 3 4 5 6

008

006

004

002

0

minus002

minus004

minus006

minus008Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at875

Hz (

mm

)

0 1 2 3 4 5 6

01

005

0

minus005

minus01Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at775

Hz (

mm

)

0 1 2 3 4 5 6

Time (s)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

0 1 2 3 4 5 6

0 1 2 3 4 5 6

0

008

006

004

002

minus002

minus004

minus006

minus008Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at895

Hz (

mm

)

02

01

0

minus01

minus02

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at835

Hz (

mm

)Figure 12 Theoretical tip displacement responses with third-type mode shape of the beam from transient to steady state

Table 3 The first natural frequencies and the resonant displacements at some certain positions

Resonant frequency or positions The values of frequencies and displacement responses (Hz or mm) ErrorsTheory based on 119880(2)

1(119909) Theory based on 119880(3)

1(119909) Experiment

The 1st natural frequency 815 815 812 037119909 = 364mm 01285 01285 01277 062119909 = 574mm 03022 03022 03001 069119909 = 784mm 05279 05279 05246 063119909 = 994mm 07905 07905 07855 063119909 = 1204mm 10742 10742 10671 066

base excitation using forced vibration theory and the methodof separation of variables A simplified cantilever model byreplacing the tip-mass with an equivalent inertial force andinertial moment acting at the free end of the cantilever beamwas detailed In addition using the Dirac Delta Functionthe base transverse excitation the concentrated equivalentinertia force and inertia moment were represented in theform of distributed loads which can make the method ofseparation of variables be utilized properly In this case sometypical results of natural frequencies and mode shapes of thecantilever beam with a tip-mass were derived using the rein-forced method of separation of variables and the developedforced vibration respectively Furthermore based on modalanalysis method the transient and steady displacement

responses of the cantilever system with a tip-mass and a sinebase excitation were presented in which two different types ofmode shapes calculated by two improved analytical methodswere used Finally an experimental cantilever system wasfabricated and some practical displacement responses weremeasured

From the results of the typical natural frequencies andmode shapes we can easily found that the mode shapesfrom the first improved type were identical to the modeshapes from the second type whether the mode shapes fromthe lower order or from the higher order even the offsetdistance was large The comparing results of the transientand steady displacement responses from the theoreticalanalysis and experimental measurements further proved the

14 Shock and Vibration

005

0

minus0050 2 4 6 8 10 12

0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12

0 2 4 6 8 10 120 2 4 6 8 10 12

0 2 4 6 8 10 12 0 2 4 6 8 10 120 2 4 6 8 10 12

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at732

Hz (

mm

)

0

008

006

004

002

minus002

minus004

minus006

minus008

0

008

006

004

002

minus002

minus004

minus006

minus008

0

008

006

004

002

minus002

minus004

minus006

minus008

0

015

01

005

minus005

minus01

01

005

0

minus005

minus01

2

15

1

05

0

minus05

minus1

minus15

minus2

03

02

01

0

minus01

minus02

03

02

01

0

minus01

minus02

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at752

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at772

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at792

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at812

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at832

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at852

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at872

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at892

Hz (

mm

)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Figure 13 Experimental displacement responses at the position of 119909 = 1204mm from transient to steady state

12

1

08

06

04

02

0120 100 80 60 40 20

78

80

82

84Theo

retic

al st

eady

disp

lace

men

t (m

m)

Position at the beam (mm)Freq

uency

(Hz)

Figure 14 Theoretical steady-displacement responses versus posi-tions and frequencies with mode 119880(2)

1(119909)

improvements of the analytical methods for calculating themode shapes of the cantilever beam with a tip-mass anda base excitation In addition the amplitudes of the modeshapes decreased with the increasing mass value of the tip-mass when the moment of inertia and offset were constantsThe amplitude of the first-order mode shape increased but

12

1

08

06

04

02

0120 100 80 60 40 20

78

80

82

84Theo

retic

al st

eady

disp

lace

men

t (m

m)

Position at the beam (mm)Freq

uency

(Hz)

Figure 15 Theoretical steady-displacement responses versus posi-tions and frequencies with mode 119880(3)

1(119909)

the other nine mode shapes decreased with the increasingmoment of inertia when the mass and offset were set asconstants Conversely the amplitude of the first mode shapedecreased but the other ninemode shapes increased with theincreasing offset distance under the condition that the massand moment of inertia were constants Globally the effect

Shock and Vibration 15

12

1

08

06

04

02

0

Expe

rimen

tal s

tead

y di

spla

cem

ent (

mm

)

120 100 80 60 40 20Position at the beam (mm)

78

80

82

84Freq

uency

(Hz)

Figure 16 Experimental steady-displacement responses versuspositions and frequencies

from varying combinations of the mass and moment inertiaof the tip-mass and the offset distance for the lower-ordermode shapes was stronger than for the higher ones

Furthermore it is clearly noticed that the mass of tip-mass can reduce the values of the natural frequencies andthe increasing mass value gave the decreasing frequencieswhen the moment of inertia of the tip-mass and the offsetdistance were set as constants Similarly the moment ofinertia can further decrease the natural frequencies and thehigher moment of inertia resulted in the lower frequencieswhen the mass and the offset were constants Differently theoffset distance can reduce the first-order natural frequenciesbut increase the other nine order frequencies and the higheroffset distance led to the lower first-order natural frequenciesbut produced the higher other nine order frequencies whenthe mass and the moment of inertia were set to constant Inglobal view the influence from the varying mass momentof inertia and offset distance for the lower-order naturalfrequencies was larger than for the higher-order ones weatherthis influence was negative or positive

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the Natural Science Foundationsof China (no 51275380) and the Scientific and TechnologicalProject of Shaanxi Province (no 2012K06-36) The authorswould like to thank the reviewers for their valuable commentson this paper

References

[1] P A A Laura J L Pombo and E A Susemihl ldquoA note on thevibrations of a clamped-free beam with a mass at the free endrdquoJournal of Sound and Vibration vol 37 no 2 pp 161ndash168 1974

[2] B Rama Bhat andHWagner ldquoNatural frequencies of a uniformcantilever with a tip mass slender in the axial directionrdquo Journalof Sound and Vibration vol 45 no 2 pp 304ndash307 1976

[3] C W S To ldquoVibration of a cantilever beam with a baseexcitation and tip massrdquo Journal of Sound and Vibration vol83 no 4 pp 445ndash460 1982

[4] P A A Laura ldquoComments on lsquoVibration of a cantilever beamwith a base excitation and tip massrsquordquo Journal of Sound andVibration vol 88 no 4 p 569 1983

[5] C W S To ldquoDiscussion on lsquovibration of a cantilever beam witha base excitation and tip massrsquordquo Journal of Sound and Vibrationvol 97 no 2 pp 349ndash351 1984

[6] R G Jacquot ldquoFurther comments on lsquoVibration of a cantileverbeam with a base excitation and tip massrsquordquo Journal of Sound andVibration vol 93 no 2 pp 312ndash313 1984

[7] S AMousavi Lajimi andG R Heppler ldquoComments on lsquonaturalfrequencies of a uniform cantilever with a tip mass slender inthe axial directionrsquordquo Journal of Sound and Vibration vol 331 no12 pp 2964ndash2968 2012

[8] R B Bhat ldquoAuthors response to the comments on lsquonaturalfrequencies of a uniform cantilever with a tip mass slender inthe axial directionrsquordquo Journal of Sound and Vibration vol 331 no12 pp 2962ndash2963 2012

[9] E Esmailzadeh and G Nakhaie-Jazar ldquoPeriodic behavior ofa cantilever beam with end mass subjected to harmonic baseexcitationrdquo International Journal of Non-Linear Mechanics vol33 no 4 pp 567ndash577 1998

[10] H Abramovich and O Hamburger ldquoVibration of a cantilevertimoshenko beam with a tip massrdquo Journal of Sound andVibration vol 148 no 1 pp 162ndash170 1991

[11] B Pratiher ldquoVibration control of a transversely excited can-tilever beam with tip massrdquo Archive of Applied Mechanics vol82 no 1 pp 31ndash42 2012

[12] M Kim M Hoegen J Dugundji and B L Wardle ldquoModelingand experimental verification of proof mass effects on vibrationenergy harvester performancerdquo Smart Materials and Structuresvol 19 Article ID 045023 2010

[13] Y T Li and R Wong ldquoIntegral and series representations ofthe Dirac delta functionrdquo Communications on Pure and AppliedAnalysis vol 7 no 2 pp 229ndash247 2008

[14] M Al-Gwaiz Sturm-LiouvilleTheory and its Applications chap-ter 3 Springer London UK 2008

[15] F Bao-lian ldquoGeneralized reciprocal theorems and their appli-cationsrdquo Applied Mathematics and Mechanics vol 23 no 2 pp203ndash210 2001

[16] C Cai H Zheng M Khan and K HungModeling of MaterialDamping Properties in ANSYS Defense Systems Division Insti-tute of High Performance Computing 2002

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Shock and Vibration

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DistributedSensor Networks

International Journal of

6 Shock and Vibration

Table 1 Typical roots of characteristic equation with variables 120601 119896119871 and 119889119871

Sample 120601 119870119871 119889119871

Roots of characteristic equation1199061

1199062

1199063

1199064

1199065

1 4 4 0 035211 129234 477818 788401 11017802 4 4 05 034814 130637 477952 788472 11018273 4 4 1 034200 132825 478225 788634 11019404 4 4 2 032544 138983 479177 789233 11023685 4 4 4 028806 154543 482543 791455 1103990

003002001

0

0 05 1 0 05 1 0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

000000

050000

100000

i = 1

000000

050000

100000

i = 1

000000 000000

000000

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

000000

050000

100000

i = 2

000000

050000

100000

i = 2

minus186571 times 10minus9

000792

003152

minus198560 times 10minus8

minus9418658

minus18545378

0

minus100

minus2000 05 1

0 05 1

0 05 1

0 05 1

1050

times104 i = 3

000000

050000

100000

i = 3

000000

050000

100000

i = 3

786191 times 10minus6

111026 times 104

minus686262 times 103

50

minus5

times105 i = 4000000

050000

100000

i = 4

000000

050000

100000

i = 4

i = 5000000

050000

100000

i = 5

000000 000000

050000

100000

i = 5times106

20

minus2

000017

minus271300 times 106

minus855944 times 104

1

05

0

1

05

0

03

02

01

0

03

02

01

0

151

050

151

050

2

0

minus2

2

0

minus2

210

minus1

210

minus1

minus493414 times 10minus8

032943

131050

367824 times 10minus11

017448

034354

113682 times 10minus10

160541

minus009923

457130 times 10minus11

minus002114

006144

866744 times 10minus11

minus140614

minus004436

032943

131050

017448

034354

160541

minus009923

000000

minus002114

006144

minus140614

minus004436

Notes at xL Notes at xL Notes at xLValues ValuesValues

Ui(x)

xL

Ui(x)

xL

Ui(x)

xL

minus233509 times 10minus5

107963 times 104

minus313859 times 104

First-type mode shapes U(1)i

(x) Second-type mode shapes U(2)i

(x) Third-type mode shapes U(3)i

(x)

Figure 4 Three types of the first five mode shapes with the combinations of 120601 = 4 119896119871 = 4 and 119889119871 = 0

It is obvious that when 119889119871 = 0 the first type of naturemode shapes 119880(1)

119894(119909) derived by the undeveloped method

turned far away from the third-type mode shapes 119880(3)119894(119909)

derived by the reinforced method of separation of variablesIn addition the larger value of 119889119871 resulted in the moreinconsistent mode shapes However the second type of modeshapes 119880(2)

119894(119909) obtained by the developed method of forced

vibration seemed in good agreement with the third onesFurthermore the amplitude of the first mode shape

decreased but the other four mode shapes increased with theincreasing value of the 119889119871 when 119896119871 and 120601 were fixed to be

constants The effect from the varying value of 119889119871 for thelower mode shapes was always stronger than for the higher

4 Experimental Verification

In order to experimentally explain and compare two typesof natural mode shapes derived by two improved methodsabove the transient and steady responses of the cantileversystem under an input base excitation should be calculatedfirst This whole calculation can be finished by method ofmodal analysis But above all the orthogonality property of

Shock and Vibration 7

0 05 1

0 05 1

Ui(x)

xL

Ui(x)

xL

Ui(x)

xL

014013012011

0

minus100

minus200

000000

050000

100000

i = 1

000000

050000

100000

i = 2

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

011044

011840

014143

533123

minus9287788

minus18666064

0 05 1

1050

times104

000000

050000

100000

i = 3minus8449356

107519 times 105

minus970528 times 103

0 05 1

50

minus5

times105

000000

050000

100000

i = 422493344

139452 times 104

139108 times 105

0 05 1

40

minus4minus8

times106

000000

050000

100000

i = 5minus42806116

minus266045 times 106

minus765307 times 106

0 05 1

000000

050000

100000

i = 106

04

02

0

392120 times 10minus9

017490

069258

0 05 1

06

04

02

0

0 05 1

000000

050000

100000

i = 2547973 times 10minus11

017881

034942

03

02

01

00 05 1

03

02

01

0

0 05 1

000000

050000

100000

i = 3minus505260 times 10minus11

160517

minus010133

151

050

0 05 1

151

050

0 05 1

000000

050000

100000

i = 4minus776048 times 10minus11

minus002162

006261

2

0

minus20 05 1

2

0

minus2

0 05 1

000000

050000

100000

i = 5663249 times 10minus11

minus140614

minus004516

210

minus1

000000

000000

000000

000000

017490

069258

017881

034942

160517

minus010133

minus002162

006261

000000

050000

100000

i = 5

0 05 1

210

minus1

000000

minus140614

minus004516

Notes at xL Notes at xL Notes at xLValues ValuesValuesFirst-type mode shapes U(1)i

(x) Second-type mode shapes U(2)i

(x) Third-type mode shapes U(3)i

(x)

Figure 5 Three types of the first five mode shapes with the combinations of 120601 = 4 119896119871 = 4 and 119889119871 = 05

034

033

032

031

0 05 1 0 05 1 0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1 0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0

minus100

minus200

10

5

0

times104

50

minus5

times105

times106

0minus5minus10minus15

Ui(x)

xL

Ui(x)

xL

Ui(x)

xL

04

02

0

04

02

0

04

02

0

04

02

0

151

050

151

050

2

0

minus2

2

0

minus2

210

minus1

210

minus1

000000

050000

100000

i = 1

000000 000000

050000

100000

i = 1

030899

031712

033934

000000

050000

100000

i = 2861965

minus9105604

minus18440224

000000

050000

100000

i = 3minus10637851

973291 times 104

minus123754 times 104

000000

050000

100000

i = 419388076

168672 times 104

311790 times 105

000000

050000

100000

i = 5minus19120922

minus244086 times 106

minus152554 times 107

000000

050000

100000

i = 1

331605 times 10minus9

012125

047836

000000

050000

100000

i = 2

minus550109 times 10minus11

018658

036219

000000

050000

100000

i = 3

minus175471 times 10minus11

160459

minus010631

000000

050000

100000

i = 4

109777 times 10minus10

minus002275

006562

000000

050000

100000

i = 5

836395 times 10minus11

minus140614

minus004731

012125

047836

000000 000000

050000

100000

i = 2

018658

036219

000000 000000

050000

100000

i = 4

minus002275

006562

000000 000000

050000

100000

i = 5

minus140614

minus004731

000000 000000

050000

100000

i = 3

160459

minus010631

Notes at xL Notes at xL Notes at xLValues ValuesValuesFirst-type mode shapes U(1)i

(x) Second-type mode shapes U(2)i

(x) Third-type mode shapes U(3)i

(x)

Figure 6 Three types of the first five mode shapes with the combinations of 120601 = 4 119896119871 = 4 and 119889119871 = 1

8 Shock and Vibration

088087086085

0 05 1 0 05 1 0 05 1

0 05 1 0 05 1

0 05 10 05 1

0 05 1 0 05 1

0 05 1 0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0

minus50

minus100

minus150

6420

minus2

times104

840

minus4

times105

0minus1minus2minus3

times107

0302010

03

02

01

0

04

02

0

04

02

0

151

050

151

050

210

minus1

210

minus1

210

minus1

210

minus1

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

084906

085736

087731

minus1308377

minus9308045

minus16072843

63490847

565642 times 104

minus172997 times 104

minus329762092

230021 times 104

667852 times 105

946242416

minus149912 times 106

minus306922 times 107

minus995048 times 10minus10

007811

030669

511030 times 10minus11

021118

040600

minus114494 times 10minus10

160244

minus012466

688252 times 10minus11

minus002686

007708

183045 times 10minus11

minus140613

minus005561

000000

000000

000000

000000

000000

007811

030669

021118

040600

160244

minus012466

minus002686

007708

minus140613

minus005561

Ui(x)

xL

Ui(x)

xL

Ui(x)

xL

Notes at xL Notes at xL Notes at xLValues ValuesValuesFirst-type mode shapes U(1)i

(x) Second-type mode shapes U(2)i

(x) Third-type mode shapes U(3)i

(x)

Figure 7 Three types of the first five mode shapes with the combinations of 120601 = 4 119896119871 = 4 and 119889119871 = 2

the natural mode shapes should be derived In fact usingBetti theorem [15] the orthogonality properties with respectto mass and stiffness can be obtained respectively as follows

int

119871

0

119898119880(119896)

119903(119909)119880(119896)

119894(119909) 119889119909 +119872119880

(119896)

119903(119871)119880(119896)

119894(119871)

+ 119903119888119872119880(119896)

119894(119871)

119889119880(119896)

119903(119871)

119889119909

+ 119903119888119872119880(119896)

119903(119871)

119889119880(119896)

119894(119871)

119889119909

+ (119869 + 1199032

119888119872)

119889119880(119896)

119894(119871)

119889119909

119889119880(119896)

119903(119871)

119889119909

= (120575(119896)

119903119894)

2

(119896 = 1 2 119903 119894 = 1 2 )

(34)

int

119871

0

119864119868

1198892119880(119896)

119894(119909)

1198891199092

1198892119880(119896)

119903(119909)

1198891199092

119889119909 = (120596119903120575(119896)

119903119894)

2

(119896 = 1 2 119903 119894 = 1 2 )

(35)

In this case the normalized mode shapes can be defined as

119880(119896)

119903119873(119909) =

119880(119896)

119903(119909)

120575(119896)

119903119903

(36)

where

(120575(119896)

119903119903)

2

= int

119871

0

119898[119880(119896)

119903(119909)]

2

119889119909 +119872[119880(119896)

119903(119871)]

2

+ 2119903119888119872119880(119896)

119903(119871) [

119889

119889119909

119880(119896)

119903(119871)]

+ (119869 +1198721199032

119888) [

119889

119889119909

119880(119896)

119903(119871)]

2

(37)

41 Theoretical Responses of the Cantilever System with a BaseSine Excitation According to the method of modal analysisthe solution of (8) has the form of

119906119886(119909 119905) =

infin

sum

119903=1

119880(119896)

119903119873(119909) 119902119903(119905) (38)

So inserting (38) into (8) gives

119864119868

infin

sum

119903=1

1198894119880(119896)

119903119873(119909)

1198891199094

119902119903(119905) + 119898

infin

sum

119903=1

119880(119896)

119903119873(119909)

1198892119902119903(119905)

1198891199052

= [119872120575 (119909 minus 119871) + 119903119888119872

119889120575 (119909 minus 119871)

119889119909

minus 119898]

1198892119887 (119905)

1198891199052

(39)

Considering the orthogonality properties of equations (34)and (35) let (39) be multiplied by119880(119896)

119894119873(119909) and integrated over

Shock and Vibration 9

192

191

19

0 05 1 0 05 1

0 05 1 0 05 1

0 05 1

0 05 10 05 1

0 05 10 05 1

0 05 1 0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0

minus200

minus400

minus600

0minus5minus10

times104

1

0

minus1

times106

times107

0minus2minus4minus6

02

01

0

02

01

0

06

04

02

0

06

04

02

0

15

1

05

0

15

1

05

0

210

minus1

210

minus1

210

minus1

210

minus1

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

000000

050000

100000

i = 1

000000

050000

100000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

189913

190621

192051

minus56420429

minus28138349

4723398

134513 times 104

minus110859 times 105

minus260952 times 104

minus615688 times 104

496921 times 104

145091 times 106

168417 times 105

241806 times 106

minus633899 times 107

137503 times 10minus9

004964

019458

minus238241 times 10minus11

028530

054402

159443

minus019150

minus283947 times 10minus11

minus004212

012050

771494 times 10minus11

minus140605

minus008755

000000

000000

000000 000000

000000

000000

004964

019458

028530

054402

159443

minus019150

minus004212

012050

minus140605

minus008755

Ui(x)

xL

Ui(x)

xL

Ui(x)

xL

Notes at xL Notes at xL Notes at xLValues ValuesValues

minus387890 times 10minus11

First-type mode shapes U(1)i

(x) Second-type mode shapes U(2)i

(x) Third-type mode shapes U(3)i

(x)

Figure 8 Three types of the first five mode shapes with the combinations of 120601 = 4 119896119871 = 4 and 119889119871 = 4

the domain 0 lt 119909 lt 119871 to give a set of independent ordinarydifferential equations as

1198892119902119894(119905)

1198891199052

+ 1205962

119894119902119894(119905) = 119891

(119896)

119894(119905) (40)

Considering the properties of Dirac Delta Function

int

119871

0

119880(119896)

119894119873(119909) 120575 (119909 minus 119871) 119889119909 = 119880

(119896)

119894119873(119871)

int

119871

0

119880(119896)

119894119873(119909)

119889120575 (119909 minus 119871)

119889119909

119889119909 = minus

119889119880(119896)

119894119873(119871)

119889119909

(41)

the generalized forces associated with the generalized coordi-nates 119902

119894(119905) can be obtained as

119891(119896)

119894(119905)

= [119872119880(119896)

119894119873(119871) minus 119903

119888119872

119889119880(119896)

119894119873(119871)

119889119909

minus 119898int

119871

0

119880(119896)

119894119873(119909) 119889119909]

1198892119887 (119905)

1198891199052

(42)

Furthermore considering the mechanical damping asRayleigh damping and ignoring the part of mass dampingthe damping coefficient 120577 of the bimorph beam can be deter-mined by the modal damping ratio which can be obtained

by experiment [16] The independent ordinary differentialequations with damping then can be written as

1198892119902119894(119905)

1198891199052

+ 2120596119894120577

119889119902119894(119905)

119889119905

+ 1205962

119894119902119894(119905)

= [119872119880(119896)

119894119873(119871) minus 119903

119888119872

119889119880(119896)

119894119873(119871)

119889119909

minus 119898int

119871

0

119880(119896)

119894119873(119909) 119889119909]

1198892119887 (119905)

1198891199052

(43)

Now (43) can be solved using Laplace transform Applyinga sine excitation 119887(119905) = 119861 sin(120596119905) to the base of cantileverstructure under zero initial conditions the transfer functionof the independent differential equations for the cantileverbeam with tip-mass under base displacement excitation canbe written as

119866119894(119904)

=

119876119894(119904)

119861119894(119904)

=

(minus1205962)[119872119880

(119896)

119894119873(119871)minus119903

119888119872(119889119880

(119896)

119894119873(119871) 119889119909)minus119898int

119871

0119880(119896)

119894119873(119909) 119889119909]

1199042+ 2120596119894120577119904 + 120596

2

119894

(44)

10 Shock and Vibration

where 119876119894(119904) and 119861

119894(119904) are the Laplace transforms of 119902

119894(119905) and

119887(119905) respectively The inverse transformation of the transferfunction will be further obtained as

119892119894(119905)

=

(minus1205962)[119872119880

(119896)

119894119873(119871)minus119903

119888119872(119889119880

(119896)

119894119873(119871) 119889119909)minus119898int

119871

0119880(119896)

119894119873(119909) 119889119909]

120596119894radic1 minus 120577

2

times 119890minus120596119894120577119905 sin(120596

119894radic1 minus 120577

2119905)

(45)

In this case the solutions of (43) can be solved by convolutiontheorem as

119902119894(119905) = 119887 (119905) lowast 119892

119894(119905) = int

119905

0

119887 (120591) 119892119894(119905 minus 120591) 119889120591 (46)

Finally substituting the result of (46) into (38) and after somelengthy algebraic manipulation the absolute displacementresponse of the cantilever beam with tip-mass under basetransverse excitation can be obtained as

119906119886(119909 120596 119905) =

infin

sum

119894=1

119880(119896)

119894119873(119909)

119861 [119872119880(119896)

119894119873(119871) minus 119903

119888119872(119889119880

(119896)

119894119873(119871) 119889119909) minus 119898int

119871

0119880(119896)

119894119873(119909) 119889119909]

radic1205964+ 2 (2120577

2minus 1) 120596

21205962

119894+ 1205964

119894

times [

1205963

120596119894radic1 minus 120577

2

119890minus120596119894120577119905 sin(120596

119894radic1 minus 120577

2119905 + 1206011) + 1205962 sin (120596119905 + 120601

2)]

(47)

where 1206011and 120601

2are determined by tan(120601

1) = 2120577radic1 minus 120577

2

((120596120596119894)2+ 21205772minus 1) and tan(120601

2) = minus2120577(120596120596

119894)(1 minus (120596120596

119894)2)

It can be seen that the solution consists of two parts abouttimeThe first term represents free vibration which will decaywith time exponentially The second term depending on theexcitation represents the forced vibration of the system

42 Experimental Responses and Comparisons with the Theo-retical Results In order to explain the absolute displacementresponse of the cantilever structure under base excitationexperimentally a typical cantilever beam in which theelongated beam was inserted and fixed into the tip-mass tomake sure the gravity center of tip-mass was collinear withthe beam was fabricated (as shown in Figure 9) The chosengeometric and physical properties of the cantilever beamincluding the tip-mass are given in Table 2

The complete experimental devices are shown inFigure 10 The electromagnetic vibrating table type SW-TFAcontrolled by a programmable controller type SW-F-2TF-2was used to create a specific harmonic input excitationThe cantilever beam with a tip-mass was clamped onthe vibrating top platform Eddy-Current Sensors typeDM6008 and ZA210800 were used to measure the inputdisplacement from base vibrating exciter and the absolutedynamic displacement from the beam respectively Allmeasurements from cantilever beam and vibrating exciterwere connected to Digital Oscilloscope type DS1204Band the Industrial Computer including a PC-based DataAcquisitionModule from ADLINK company type DAQ2213The Digital Oscilloscope and Industrial Computer displayedand processed the measurement results using the DAQPilotsoftware and the results from subsequent analysis using theMATLAB software

Table 2 Characteristic properties of the cantilever beam includingtip-mass

Physical properties Beam Tip-massYoungrsquos modulus 119864 (GPa) 69 mdashDensity 120588

119904or 120588119898(kgm3) 2630 7753

Length 119871 or 119871119898(mm) 1204 156

Thickness 119905119904or 119905119898(mm) 49 295

Width 119908 (mm) 102 102Total mass119872 (g)a mdash 3383Offset 119903

119888(mm)b mdash 801

Rotary inertia 119869 (kgsdotm2)b c mdash 333 times 10minus6aThe total mass of the tip-mass including the elongated part of beam wascalculated by119872 = 119871

119898119905119898119908120588119898+ 119871119904119905119904119908(120588119904minus 120588119898) where 119871

119904= 10mm

bThe offset distance and the rotary inertia were calculated according to thegeometry and material properties of the beam and tip-mass using equations119903119888= (120588

1198981199051198981198712

119898minus 1205881198981199051199041198712

119904+ 1205881199041199051199041198712

119904)(2[120588

119898119905119898119871119898+ 119871119904119905119904(120588119904minus 120588119898)])

119869 = 120588119898119871119898119905119898119908 [(112)(119871

2

119898+ 1199052

119898) + ((12)119871

119898minus 119903119888)2] + 119871

119904119905119904119908(120588119904minus

120588119898)[(112)(119871

2

119904+ 1199052

119904) + (119903119888minus (12)119871

119904)2]

cThe corresponding parameters 120601 119896119871 and 119903119888119871 were calculated as follows

120601 = 119872(120588119904119905119904119908) = 21377 119896119871 = radic119869119872119871 = 00824 and 119903

119888119871 = 00665

The analytical model defined by (47) in which two typesof mode shapes derived by two improved analytical methodsare utilized can be used to analyze the steady and transientdisplacement responses of the cantilever beam with a tip-mass and an input base excitation based on the chosenvarying frequencies time and positions along the lengthof the beam Typically the displacement responses frompositions along 119909-axis at 119909

1= 364mm 119909

2= 574mm

1199093

= 784mm 1199094

= 994mm and 1199095

= 1204mmoff the origin point (as shown in Figure 9) were measured

Shock and Vibration 11

y

b(t)

x1

x2

x3

x4

x5

Ls

L

t s

t m x

Lm

rc

Cg

o998400

Figure 9 Typical cantilever beam with a tip-mass

1 2

3

56

7

8

4

Tip-massCantilever beam

(7) Cantilever with a tip-mass

(1) Industrial computer including data acquisitionmodule type DAQ2213

(2) Digital oscilloscope type DS1204B(3) Programmable controller type SW-F-FTF-2(4) Power supply type DP1308A(5) Driver modules for Eddy-current sensors(6) Electromagnetic vibrating table type SW-TFA

(8) Eddy-current sensors type DM6008 and ZA210800

Figure 10 Complete experimental setup

by Eddy-Current Sensors The amplitude of the input sine-displacement excitation for analysis was measured as 119861 =

001mm and the damping ratio for the transverse vibrationaround the fundamental resonant frequency was found to be120577 = 00025 by Logarithmic Decrement Method

In the following paragraphs we focus on the transientand steady responses of the cantilever system with a sine baseexcitation

Figures 11 and 12 show the theoretical displacementresponses at the free end of the beam from transient tosteady state with two improved mode shapes above at thefirst resonant frequency It can be seen obviously that thedisplacement responses with two improved mode shapeswe proposed are almost identical This richly proves thatthe first-type mode shapes reformulated by the reinforcedmethod of separation of variables are equal to the second-type mode shapes derived by the developed method offorced vibration In addition it can be further obtainedthat when the frequency from the base excitation is veryclose but not exactly equal to the resonant frequency of freevibrations of the cantilever system a kind of vibration calledbeating occurs and the period of the beating increases as120596 approached 120596

1 This kind of beating decays exponentially

along the time due to the mechanical dampingFigure 13 shows the experimental displacement responses

at the position of 119909 = 1204mm from transient to steady

state As the same to the theoretical results experimental dis-placement response can be regarded as the sum of waveformsfrom transient and steady states But the transient responsevanished gradually with the time due to the mechanicaldamping and only the steady response left In addition thephenomenon of beating still existed The amplitude of thisbeating phenomenon reduced gradually and the closer theexciting frequency approached to the first natural frequencyof the cantilever system the faster the amplitude of thebeating decreased

Furthermore the amplitudes of the theoreticalwaveformsin Figures 11 and 12 are different from the experimental resultin Figure 13 This may because af the following (a) The effectof mechanical damping with respect to stiffness was onlyconsidered and the mass damping was ignored in theoreticalanalysis which can also weaken the transient vibration faster(b) The first natural frequency of the cantilever system wascalculated and measured by theoretical and experimentalmethods as 815Hz and 812Hz respectively The little dif-ference made that the max amplitudes from theoretical andexperimental results did not appear at the same frequency(c) In theoretical analysis the tip-mass was equated by theinertia force and inertia moment acting at the free end ofthe cantilever so the total length for theoretical analysiswas just the length of the beam of 1204mm Howeverduring experiments the tip-displacement was measured at

12 Shock and Vibration

005

0

minus0050 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6

0 1 2 3 4 5 6 0 1 2 3 4 5 6

0 1 2 3 4 5 6 0 1 2 3 4 5 6

0 1 2 3 4 5 6

0

0

1 2 3 4 5 6

02

01

0

minus01

minus02

02

01

0

0

minus01

minus02

015

01

005

minus005

minus01

008

006

004

002

0

minus002

minus004

minus006

minus008

008

006

004

002

0

minus002

minus004

minus006

minus008

008

006

004

002

minus002

minus004

minus006

minus008

2

15

1

05

0

minus05

minus1

minus15

minus2

01

005

0

minus005

minus01Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at735

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

795

Hz (

mm

)

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at855

Hz (

mm

)

tran

sient

to st

eady

at755

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mTh

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

815

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

875

Hz (

mm

)

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at775

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

835

Hz (

mm

)

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at895

Hz (

mm

)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Figure 11 Theoretical tip displacement responses with the second-type mode shape of the beam from transient to steady state

the approximate position of 119909 = 1204mm This is justan approximate value because Eddy-Current Sensor cannotgive exact measurement results of the displacement at thejunction between the cantilever and the tip-mass This maycause that the amplitude level from theoretical discussionwashigher than from experimental measurement In additionthe amplitude of the instantaneous base displacement fromthe exciter may be not the fixed value 001mm at the startinginstant Therefore the maximum displacement responsesduring the transient state in the experimental measurementsmay shift away from the real experimental results

The theoretical responses with two improved types ofmode shapes and the experimental displacement responsesversus varying positions of beam and the frequencies atsteady state are shown in Figures 14ndash16 respectively Figures14 and 15 show two types of theoretical displacements atthe positions of 119909 = 364mm 574mm 784mm 994mmand 1204mmwith continuous varying frequencies In exper-iments the displacement responses were measured at thesame positions with theoretical calculations but with somecertain frequencies of 782Hz 792Hz 802Hz 807Hz81Hz 812Hz 817Hz 82Hz 83Hz and 84Hz as plottedin Figure 16 It is noticed that when the frequency of baseexcitation approached to the first natural frequency of thecantilever system the amplitude of forced vibration willrapidly increase and become maximum for the case when

the frequency of the excitation coincides with the naturalfrequency Table 3 lists the values of the first natural frequen-cies and the displacement responses from theoretical andexperimental results at some certain positions long the beamunder the first natural frequencies It can be seen that thetheoretical and experimental results are matched very close

Table 3 The first natural frequencies and the resonantdisplacements at some certain positions

In a word the theoretical results with two different modeshapes derived by two improved analytical methods haveextremely the same behaviors at transient and steady stateswith respect to varying frequencies from base excitationThedisplacement responses indicate close agreement betweenthe experimental and analytical results All of these resultsgive full verification for the improvement of the analyticalmethods in this paper utilized in recalculating the modeshapes of a cantilever beam when an offset distance existsbetween the gravity center of the tip-mass and the point ofattachment

5 Conclusions

This paper presents two improved analytical methods ofcalculation for natural frequencies and mode shapes of auniform cantilever beam carrying a tip-mass under an input

Shock and Vibration 13

005

0

minus005Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at735

Hz (

mm

)

0 1 2 3 4 5 6

Time (s)

0 1 2 3 4 5 6

0 1 2 3 4 5 6

02

01

0

minus01

minus02

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at795

Hz (

mm

)

0

015

01

005

minus005

minus01

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at855

Hz (

mm

)

0 1 2 3 4 5 6

Time (s)

008

006

004

002

0

minus002

minus004

minus006

minus008tran

sient

to st

eady

at755

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

m

2

15

1

05

0

minus05

minus1

minus15

minus2Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at815

Hz (

mm

)

0 1 2 3 4 5 6

008

006

004

002

0

minus002

minus004

minus006

minus008Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at875

Hz (

mm

)

0 1 2 3 4 5 6

01

005

0

minus005

minus01Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at775

Hz (

mm

)

0 1 2 3 4 5 6

Time (s)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

0 1 2 3 4 5 6

0 1 2 3 4 5 6

0

008

006

004

002

minus002

minus004

minus006

minus008Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at895

Hz (

mm

)

02

01

0

minus01

minus02

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at835

Hz (

mm

)Figure 12 Theoretical tip displacement responses with third-type mode shape of the beam from transient to steady state

Table 3 The first natural frequencies and the resonant displacements at some certain positions

Resonant frequency or positions The values of frequencies and displacement responses (Hz or mm) ErrorsTheory based on 119880(2)

1(119909) Theory based on 119880(3)

1(119909) Experiment

The 1st natural frequency 815 815 812 037119909 = 364mm 01285 01285 01277 062119909 = 574mm 03022 03022 03001 069119909 = 784mm 05279 05279 05246 063119909 = 994mm 07905 07905 07855 063119909 = 1204mm 10742 10742 10671 066

base excitation using forced vibration theory and the methodof separation of variables A simplified cantilever model byreplacing the tip-mass with an equivalent inertial force andinertial moment acting at the free end of the cantilever beamwas detailed In addition using the Dirac Delta Functionthe base transverse excitation the concentrated equivalentinertia force and inertia moment were represented in theform of distributed loads which can make the method ofseparation of variables be utilized properly In this case sometypical results of natural frequencies and mode shapes of thecantilever beam with a tip-mass were derived using the rein-forced method of separation of variables and the developedforced vibration respectively Furthermore based on modalanalysis method the transient and steady displacement

responses of the cantilever system with a tip-mass and a sinebase excitation were presented in which two different types ofmode shapes calculated by two improved analytical methodswere used Finally an experimental cantilever system wasfabricated and some practical displacement responses weremeasured

From the results of the typical natural frequencies andmode shapes we can easily found that the mode shapesfrom the first improved type were identical to the modeshapes from the second type whether the mode shapes fromthe lower order or from the higher order even the offsetdistance was large The comparing results of the transientand steady displacement responses from the theoreticalanalysis and experimental measurements further proved the

14 Shock and Vibration

005

0

minus0050 2 4 6 8 10 12

0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12

0 2 4 6 8 10 120 2 4 6 8 10 12

0 2 4 6 8 10 12 0 2 4 6 8 10 120 2 4 6 8 10 12

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at732

Hz (

mm

)

0

008

006

004

002

minus002

minus004

minus006

minus008

0

008

006

004

002

minus002

minus004

minus006

minus008

0

008

006

004

002

minus002

minus004

minus006

minus008

0

015

01

005

minus005

minus01

01

005

0

minus005

minus01

2

15

1

05

0

minus05

minus1

minus15

minus2

03

02

01

0

minus01

minus02

03

02

01

0

minus01

minus02

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at752

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at772

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at792

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at812

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at832

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at852

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at872

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at892

Hz (

mm

)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Figure 13 Experimental displacement responses at the position of 119909 = 1204mm from transient to steady state

12

1

08

06

04

02

0120 100 80 60 40 20

78

80

82

84Theo

retic

al st

eady

disp

lace

men

t (m

m)

Position at the beam (mm)Freq

uency

(Hz)

Figure 14 Theoretical steady-displacement responses versus posi-tions and frequencies with mode 119880(2)

1(119909)

improvements of the analytical methods for calculating themode shapes of the cantilever beam with a tip-mass anda base excitation In addition the amplitudes of the modeshapes decreased with the increasing mass value of the tip-mass when the moment of inertia and offset were constantsThe amplitude of the first-order mode shape increased but

12

1

08

06

04

02

0120 100 80 60 40 20

78

80

82

84Theo

retic

al st

eady

disp

lace

men

t (m

m)

Position at the beam (mm)Freq

uency

(Hz)

Figure 15 Theoretical steady-displacement responses versus posi-tions and frequencies with mode 119880(3)

1(119909)

the other nine mode shapes decreased with the increasingmoment of inertia when the mass and offset were set asconstants Conversely the amplitude of the first mode shapedecreased but the other ninemode shapes increased with theincreasing offset distance under the condition that the massand moment of inertia were constants Globally the effect

Shock and Vibration 15

12

1

08

06

04

02

0

Expe

rimen

tal s

tead

y di

spla

cem

ent (

mm

)

120 100 80 60 40 20Position at the beam (mm)

78

80

82

84Freq

uency

(Hz)

Figure 16 Experimental steady-displacement responses versuspositions and frequencies

from varying combinations of the mass and moment inertiaof the tip-mass and the offset distance for the lower-ordermode shapes was stronger than for the higher ones

Furthermore it is clearly noticed that the mass of tip-mass can reduce the values of the natural frequencies andthe increasing mass value gave the decreasing frequencieswhen the moment of inertia of the tip-mass and the offsetdistance were set as constants Similarly the moment ofinertia can further decrease the natural frequencies and thehigher moment of inertia resulted in the lower frequencieswhen the mass and the offset were constants Differently theoffset distance can reduce the first-order natural frequenciesbut increase the other nine order frequencies and the higheroffset distance led to the lower first-order natural frequenciesbut produced the higher other nine order frequencies whenthe mass and the moment of inertia were set to constant Inglobal view the influence from the varying mass momentof inertia and offset distance for the lower-order naturalfrequencies was larger than for the higher-order ones weatherthis influence was negative or positive

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the Natural Science Foundationsof China (no 51275380) and the Scientific and TechnologicalProject of Shaanxi Province (no 2012K06-36) The authorswould like to thank the reviewers for their valuable commentson this paper

References

[1] P A A Laura J L Pombo and E A Susemihl ldquoA note on thevibrations of a clamped-free beam with a mass at the free endrdquoJournal of Sound and Vibration vol 37 no 2 pp 161ndash168 1974

[2] B Rama Bhat andHWagner ldquoNatural frequencies of a uniformcantilever with a tip mass slender in the axial directionrdquo Journalof Sound and Vibration vol 45 no 2 pp 304ndash307 1976

[3] C W S To ldquoVibration of a cantilever beam with a baseexcitation and tip massrdquo Journal of Sound and Vibration vol83 no 4 pp 445ndash460 1982

[4] P A A Laura ldquoComments on lsquoVibration of a cantilever beamwith a base excitation and tip massrsquordquo Journal of Sound andVibration vol 88 no 4 p 569 1983

[5] C W S To ldquoDiscussion on lsquovibration of a cantilever beam witha base excitation and tip massrsquordquo Journal of Sound and Vibrationvol 97 no 2 pp 349ndash351 1984

[6] R G Jacquot ldquoFurther comments on lsquoVibration of a cantileverbeam with a base excitation and tip massrsquordquo Journal of Sound andVibration vol 93 no 2 pp 312ndash313 1984

[7] S AMousavi Lajimi andG R Heppler ldquoComments on lsquonaturalfrequencies of a uniform cantilever with a tip mass slender inthe axial directionrsquordquo Journal of Sound and Vibration vol 331 no12 pp 2964ndash2968 2012

[8] R B Bhat ldquoAuthors response to the comments on lsquonaturalfrequencies of a uniform cantilever with a tip mass slender inthe axial directionrsquordquo Journal of Sound and Vibration vol 331 no12 pp 2962ndash2963 2012

[9] E Esmailzadeh and G Nakhaie-Jazar ldquoPeriodic behavior ofa cantilever beam with end mass subjected to harmonic baseexcitationrdquo International Journal of Non-Linear Mechanics vol33 no 4 pp 567ndash577 1998

[10] H Abramovich and O Hamburger ldquoVibration of a cantilevertimoshenko beam with a tip massrdquo Journal of Sound andVibration vol 148 no 1 pp 162ndash170 1991

[11] B Pratiher ldquoVibration control of a transversely excited can-tilever beam with tip massrdquo Archive of Applied Mechanics vol82 no 1 pp 31ndash42 2012

[12] M Kim M Hoegen J Dugundji and B L Wardle ldquoModelingand experimental verification of proof mass effects on vibrationenergy harvester performancerdquo Smart Materials and Structuresvol 19 Article ID 045023 2010

[13] Y T Li and R Wong ldquoIntegral and series representations ofthe Dirac delta functionrdquo Communications on Pure and AppliedAnalysis vol 7 no 2 pp 229ndash247 2008

[14] M Al-Gwaiz Sturm-LiouvilleTheory and its Applications chap-ter 3 Springer London UK 2008

[15] F Bao-lian ldquoGeneralized reciprocal theorems and their appli-cationsrdquo Applied Mathematics and Mechanics vol 23 no 2 pp203ndash210 2001

[16] C Cai H Zheng M Khan and K HungModeling of MaterialDamping Properties in ANSYS Defense Systems Division Insti-tute of High Performance Computing 2002

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Shock and Vibration

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Shock and Vibration 7

0 05 1

0 05 1

Ui(x)

xL

Ui(x)

xL

Ui(x)

xL

014013012011

0

minus100

minus200

000000

050000

100000

i = 1

000000

050000

100000

i = 2

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

011044

011840

014143

533123

minus9287788

minus18666064

0 05 1

1050

times104

000000

050000

100000

i = 3minus8449356

107519 times 105

minus970528 times 103

0 05 1

50

minus5

times105

000000

050000

100000

i = 422493344

139452 times 104

139108 times 105

0 05 1

40

minus4minus8

times106

000000

050000

100000

i = 5minus42806116

minus266045 times 106

minus765307 times 106

0 05 1

000000

050000

100000

i = 106

04

02

0

392120 times 10minus9

017490

069258

0 05 1

06

04

02

0

0 05 1

000000

050000

100000

i = 2547973 times 10minus11

017881

034942

03

02

01

00 05 1

03

02

01

0

0 05 1

000000

050000

100000

i = 3minus505260 times 10minus11

160517

minus010133

151

050

0 05 1

151

050

0 05 1

000000

050000

100000

i = 4minus776048 times 10minus11

minus002162

006261

2

0

minus20 05 1

2

0

minus2

0 05 1

000000

050000

100000

i = 5663249 times 10minus11

minus140614

minus004516

210

minus1

000000

000000

000000

000000

017490

069258

017881

034942

160517

minus010133

minus002162

006261

000000

050000

100000

i = 5

0 05 1

210

minus1

000000

minus140614

minus004516

Notes at xL Notes at xL Notes at xLValues ValuesValuesFirst-type mode shapes U(1)i

(x) Second-type mode shapes U(2)i

(x) Third-type mode shapes U(3)i

(x)

Figure 5 Three types of the first five mode shapes with the combinations of 120601 = 4 119896119871 = 4 and 119889119871 = 05

034

033

032

031

0 05 1 0 05 1 0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1 0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0

minus100

minus200

10

5

0

times104

50

minus5

times105

times106

0minus5minus10minus15

Ui(x)

xL

Ui(x)

xL

Ui(x)

xL

04

02

0

04

02

0

04

02

0

04

02

0

151

050

151

050

2

0

minus2

2

0

minus2

210

minus1

210

minus1

000000

050000

100000

i = 1

000000 000000

050000

100000

i = 1

030899

031712

033934

000000

050000

100000

i = 2861965

minus9105604

minus18440224

000000

050000

100000

i = 3minus10637851

973291 times 104

minus123754 times 104

000000

050000

100000

i = 419388076

168672 times 104

311790 times 105

000000

050000

100000

i = 5minus19120922

minus244086 times 106

minus152554 times 107

000000

050000

100000

i = 1

331605 times 10minus9

012125

047836

000000

050000

100000

i = 2

minus550109 times 10minus11

018658

036219

000000

050000

100000

i = 3

minus175471 times 10minus11

160459

minus010631

000000

050000

100000

i = 4

109777 times 10minus10

minus002275

006562

000000

050000

100000

i = 5

836395 times 10minus11

minus140614

minus004731

012125

047836

000000 000000

050000

100000

i = 2

018658

036219

000000 000000

050000

100000

i = 4

minus002275

006562

000000 000000

050000

100000

i = 5

minus140614

minus004731

000000 000000

050000

100000

i = 3

160459

minus010631

Notes at xL Notes at xL Notes at xLValues ValuesValuesFirst-type mode shapes U(1)i

(x) Second-type mode shapes U(2)i

(x) Third-type mode shapes U(3)i

(x)

Figure 6 Three types of the first five mode shapes with the combinations of 120601 = 4 119896119871 = 4 and 119889119871 = 1

8 Shock and Vibration

088087086085

0 05 1 0 05 1 0 05 1

0 05 1 0 05 1

0 05 10 05 1

0 05 1 0 05 1

0 05 1 0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0

minus50

minus100

minus150

6420

minus2

times104

840

minus4

times105

0minus1minus2minus3

times107

0302010

03

02

01

0

04

02

0

04

02

0

151

050

151

050

210

minus1

210

minus1

210

minus1

210

minus1

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

084906

085736

087731

minus1308377

minus9308045

minus16072843

63490847

565642 times 104

minus172997 times 104

minus329762092

230021 times 104

667852 times 105

946242416

minus149912 times 106

minus306922 times 107

minus995048 times 10minus10

007811

030669

511030 times 10minus11

021118

040600

minus114494 times 10minus10

160244

minus012466

688252 times 10minus11

minus002686

007708

183045 times 10minus11

minus140613

minus005561

000000

000000

000000

000000

000000

007811

030669

021118

040600

160244

minus012466

minus002686

007708

minus140613

minus005561

Ui(x)

xL

Ui(x)

xL

Ui(x)

xL

Notes at xL Notes at xL Notes at xLValues ValuesValuesFirst-type mode shapes U(1)i

(x) Second-type mode shapes U(2)i

(x) Third-type mode shapes U(3)i

(x)

Figure 7 Three types of the first five mode shapes with the combinations of 120601 = 4 119896119871 = 4 and 119889119871 = 2

the natural mode shapes should be derived In fact usingBetti theorem [15] the orthogonality properties with respectto mass and stiffness can be obtained respectively as follows

int

119871

0

119898119880(119896)

119903(119909)119880(119896)

119894(119909) 119889119909 +119872119880

(119896)

119903(119871)119880(119896)

119894(119871)

+ 119903119888119872119880(119896)

119894(119871)

119889119880(119896)

119903(119871)

119889119909

+ 119903119888119872119880(119896)

119903(119871)

119889119880(119896)

119894(119871)

119889119909

+ (119869 + 1199032

119888119872)

119889119880(119896)

119894(119871)

119889119909

119889119880(119896)

119903(119871)

119889119909

= (120575(119896)

119903119894)

2

(119896 = 1 2 119903 119894 = 1 2 )

(34)

int

119871

0

119864119868

1198892119880(119896)

119894(119909)

1198891199092

1198892119880(119896)

119903(119909)

1198891199092

119889119909 = (120596119903120575(119896)

119903119894)

2

(119896 = 1 2 119903 119894 = 1 2 )

(35)

In this case the normalized mode shapes can be defined as

119880(119896)

119903119873(119909) =

119880(119896)

119903(119909)

120575(119896)

119903119903

(36)

where

(120575(119896)

119903119903)

2

= int

119871

0

119898[119880(119896)

119903(119909)]

2

119889119909 +119872[119880(119896)

119903(119871)]

2

+ 2119903119888119872119880(119896)

119903(119871) [

119889

119889119909

119880(119896)

119903(119871)]

+ (119869 +1198721199032

119888) [

119889

119889119909

119880(119896)

119903(119871)]

2

(37)

41 Theoretical Responses of the Cantilever System with a BaseSine Excitation According to the method of modal analysisthe solution of (8) has the form of

119906119886(119909 119905) =

infin

sum

119903=1

119880(119896)

119903119873(119909) 119902119903(119905) (38)

So inserting (38) into (8) gives

119864119868

infin

sum

119903=1

1198894119880(119896)

119903119873(119909)

1198891199094

119902119903(119905) + 119898

infin

sum

119903=1

119880(119896)

119903119873(119909)

1198892119902119903(119905)

1198891199052

= [119872120575 (119909 minus 119871) + 119903119888119872

119889120575 (119909 minus 119871)

119889119909

minus 119898]

1198892119887 (119905)

1198891199052

(39)

Considering the orthogonality properties of equations (34)and (35) let (39) be multiplied by119880(119896)

119894119873(119909) and integrated over

Shock and Vibration 9

192

191

19

0 05 1 0 05 1

0 05 1 0 05 1

0 05 1

0 05 10 05 1

0 05 10 05 1

0 05 1 0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0

minus200

minus400

minus600

0minus5minus10

times104

1

0

minus1

times106

times107

0minus2minus4minus6

02

01

0

02

01

0

06

04

02

0

06

04

02

0

15

1

05

0

15

1

05

0

210

minus1

210

minus1

210

minus1

210

minus1

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

000000

050000

100000

i = 1

000000

050000

100000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

189913

190621

192051

minus56420429

minus28138349

4723398

134513 times 104

minus110859 times 105

minus260952 times 104

minus615688 times 104

496921 times 104

145091 times 106

168417 times 105

241806 times 106

minus633899 times 107

137503 times 10minus9

004964

019458

minus238241 times 10minus11

028530

054402

159443

minus019150

minus283947 times 10minus11

minus004212

012050

771494 times 10minus11

minus140605

minus008755

000000

000000

000000 000000

000000

000000

004964

019458

028530

054402

159443

minus019150

minus004212

012050

minus140605

minus008755

Ui(x)

xL

Ui(x)

xL

Ui(x)

xL

Notes at xL Notes at xL Notes at xLValues ValuesValues

minus387890 times 10minus11

First-type mode shapes U(1)i

(x) Second-type mode shapes U(2)i

(x) Third-type mode shapes U(3)i

(x)

Figure 8 Three types of the first five mode shapes with the combinations of 120601 = 4 119896119871 = 4 and 119889119871 = 4

the domain 0 lt 119909 lt 119871 to give a set of independent ordinarydifferential equations as

1198892119902119894(119905)

1198891199052

+ 1205962

119894119902119894(119905) = 119891

(119896)

119894(119905) (40)

Considering the properties of Dirac Delta Function

int

119871

0

119880(119896)

119894119873(119909) 120575 (119909 minus 119871) 119889119909 = 119880

(119896)

119894119873(119871)

int

119871

0

119880(119896)

119894119873(119909)

119889120575 (119909 minus 119871)

119889119909

119889119909 = minus

119889119880(119896)

119894119873(119871)

119889119909

(41)

the generalized forces associated with the generalized coordi-nates 119902

119894(119905) can be obtained as

119891(119896)

119894(119905)

= [119872119880(119896)

119894119873(119871) minus 119903

119888119872

119889119880(119896)

119894119873(119871)

119889119909

minus 119898int

119871

0

119880(119896)

119894119873(119909) 119889119909]

1198892119887 (119905)

1198891199052

(42)

Furthermore considering the mechanical damping asRayleigh damping and ignoring the part of mass dampingthe damping coefficient 120577 of the bimorph beam can be deter-mined by the modal damping ratio which can be obtained

by experiment [16] The independent ordinary differentialequations with damping then can be written as

1198892119902119894(119905)

1198891199052

+ 2120596119894120577

119889119902119894(119905)

119889119905

+ 1205962

119894119902119894(119905)

= [119872119880(119896)

119894119873(119871) minus 119903

119888119872

119889119880(119896)

119894119873(119871)

119889119909

minus 119898int

119871

0

119880(119896)

119894119873(119909) 119889119909]

1198892119887 (119905)

1198891199052

(43)

Now (43) can be solved using Laplace transform Applyinga sine excitation 119887(119905) = 119861 sin(120596119905) to the base of cantileverstructure under zero initial conditions the transfer functionof the independent differential equations for the cantileverbeam with tip-mass under base displacement excitation canbe written as

119866119894(119904)

=

119876119894(119904)

119861119894(119904)

=

(minus1205962)[119872119880

(119896)

119894119873(119871)minus119903

119888119872(119889119880

(119896)

119894119873(119871) 119889119909)minus119898int

119871

0119880(119896)

119894119873(119909) 119889119909]

1199042+ 2120596119894120577119904 + 120596

2

119894

(44)

10 Shock and Vibration

where 119876119894(119904) and 119861

119894(119904) are the Laplace transforms of 119902

119894(119905) and

119887(119905) respectively The inverse transformation of the transferfunction will be further obtained as

119892119894(119905)

=

(minus1205962)[119872119880

(119896)

119894119873(119871)minus119903

119888119872(119889119880

(119896)

119894119873(119871) 119889119909)minus119898int

119871

0119880(119896)

119894119873(119909) 119889119909]

120596119894radic1 minus 120577

2

times 119890minus120596119894120577119905 sin(120596

119894radic1 minus 120577

2119905)

(45)

In this case the solutions of (43) can be solved by convolutiontheorem as

119902119894(119905) = 119887 (119905) lowast 119892

119894(119905) = int

119905

0

119887 (120591) 119892119894(119905 minus 120591) 119889120591 (46)

Finally substituting the result of (46) into (38) and after somelengthy algebraic manipulation the absolute displacementresponse of the cantilever beam with tip-mass under basetransverse excitation can be obtained as

119906119886(119909 120596 119905) =

infin

sum

119894=1

119880(119896)

119894119873(119909)

119861 [119872119880(119896)

119894119873(119871) minus 119903

119888119872(119889119880

(119896)

119894119873(119871) 119889119909) minus 119898int

119871

0119880(119896)

119894119873(119909) 119889119909]

radic1205964+ 2 (2120577

2minus 1) 120596

21205962

119894+ 1205964

119894

times [

1205963

120596119894radic1 minus 120577

2

119890minus120596119894120577119905 sin(120596

119894radic1 minus 120577

2119905 + 1206011) + 1205962 sin (120596119905 + 120601

2)]

(47)

where 1206011and 120601

2are determined by tan(120601

1) = 2120577radic1 minus 120577

2

((120596120596119894)2+ 21205772minus 1) and tan(120601

2) = minus2120577(120596120596

119894)(1 minus (120596120596

119894)2)

It can be seen that the solution consists of two parts abouttimeThe first term represents free vibration which will decaywith time exponentially The second term depending on theexcitation represents the forced vibration of the system

42 Experimental Responses and Comparisons with the Theo-retical Results In order to explain the absolute displacementresponse of the cantilever structure under base excitationexperimentally a typical cantilever beam in which theelongated beam was inserted and fixed into the tip-mass tomake sure the gravity center of tip-mass was collinear withthe beam was fabricated (as shown in Figure 9) The chosengeometric and physical properties of the cantilever beamincluding the tip-mass are given in Table 2

The complete experimental devices are shown inFigure 10 The electromagnetic vibrating table type SW-TFAcontrolled by a programmable controller type SW-F-2TF-2was used to create a specific harmonic input excitationThe cantilever beam with a tip-mass was clamped onthe vibrating top platform Eddy-Current Sensors typeDM6008 and ZA210800 were used to measure the inputdisplacement from base vibrating exciter and the absolutedynamic displacement from the beam respectively Allmeasurements from cantilever beam and vibrating exciterwere connected to Digital Oscilloscope type DS1204Band the Industrial Computer including a PC-based DataAcquisitionModule from ADLINK company type DAQ2213The Digital Oscilloscope and Industrial Computer displayedand processed the measurement results using the DAQPilotsoftware and the results from subsequent analysis using theMATLAB software

Table 2 Characteristic properties of the cantilever beam includingtip-mass

Physical properties Beam Tip-massYoungrsquos modulus 119864 (GPa) 69 mdashDensity 120588

119904or 120588119898(kgm3) 2630 7753

Length 119871 or 119871119898(mm) 1204 156

Thickness 119905119904or 119905119898(mm) 49 295

Width 119908 (mm) 102 102Total mass119872 (g)a mdash 3383Offset 119903

119888(mm)b mdash 801

Rotary inertia 119869 (kgsdotm2)b c mdash 333 times 10minus6aThe total mass of the tip-mass including the elongated part of beam wascalculated by119872 = 119871

119898119905119898119908120588119898+ 119871119904119905119904119908(120588119904minus 120588119898) where 119871

119904= 10mm

bThe offset distance and the rotary inertia were calculated according to thegeometry and material properties of the beam and tip-mass using equations119903119888= (120588

1198981199051198981198712

119898minus 1205881198981199051199041198712

119904+ 1205881199041199051199041198712

119904)(2[120588

119898119905119898119871119898+ 119871119904119905119904(120588119904minus 120588119898)])

119869 = 120588119898119871119898119905119898119908 [(112)(119871

2

119898+ 1199052

119898) + ((12)119871

119898minus 119903119888)2] + 119871

119904119905119904119908(120588119904minus

120588119898)[(112)(119871

2

119904+ 1199052

119904) + (119903119888minus (12)119871

119904)2]

cThe corresponding parameters 120601 119896119871 and 119903119888119871 were calculated as follows

120601 = 119872(120588119904119905119904119908) = 21377 119896119871 = radic119869119872119871 = 00824 and 119903

119888119871 = 00665

The analytical model defined by (47) in which two typesof mode shapes derived by two improved analytical methodsare utilized can be used to analyze the steady and transientdisplacement responses of the cantilever beam with a tip-mass and an input base excitation based on the chosenvarying frequencies time and positions along the lengthof the beam Typically the displacement responses frompositions along 119909-axis at 119909

1= 364mm 119909

2= 574mm

1199093

= 784mm 1199094

= 994mm and 1199095

= 1204mmoff the origin point (as shown in Figure 9) were measured

Shock and Vibration 11

y

b(t)

x1

x2

x3

x4

x5

Ls

L

t s

t m x

Lm

rc

Cg

o998400

Figure 9 Typical cantilever beam with a tip-mass

1 2

3

56

7

8

4

Tip-massCantilever beam

(7) Cantilever with a tip-mass

(1) Industrial computer including data acquisitionmodule type DAQ2213

(2) Digital oscilloscope type DS1204B(3) Programmable controller type SW-F-FTF-2(4) Power supply type DP1308A(5) Driver modules for Eddy-current sensors(6) Electromagnetic vibrating table type SW-TFA

(8) Eddy-current sensors type DM6008 and ZA210800

Figure 10 Complete experimental setup

by Eddy-Current Sensors The amplitude of the input sine-displacement excitation for analysis was measured as 119861 =

001mm and the damping ratio for the transverse vibrationaround the fundamental resonant frequency was found to be120577 = 00025 by Logarithmic Decrement Method

In the following paragraphs we focus on the transientand steady responses of the cantilever system with a sine baseexcitation

Figures 11 and 12 show the theoretical displacementresponses at the free end of the beam from transient tosteady state with two improved mode shapes above at thefirst resonant frequency It can be seen obviously that thedisplacement responses with two improved mode shapeswe proposed are almost identical This richly proves thatthe first-type mode shapes reformulated by the reinforcedmethod of separation of variables are equal to the second-type mode shapes derived by the developed method offorced vibration In addition it can be further obtainedthat when the frequency from the base excitation is veryclose but not exactly equal to the resonant frequency of freevibrations of the cantilever system a kind of vibration calledbeating occurs and the period of the beating increases as120596 approached 120596

1 This kind of beating decays exponentially

along the time due to the mechanical dampingFigure 13 shows the experimental displacement responses

at the position of 119909 = 1204mm from transient to steady

state As the same to the theoretical results experimental dis-placement response can be regarded as the sum of waveformsfrom transient and steady states But the transient responsevanished gradually with the time due to the mechanicaldamping and only the steady response left In addition thephenomenon of beating still existed The amplitude of thisbeating phenomenon reduced gradually and the closer theexciting frequency approached to the first natural frequencyof the cantilever system the faster the amplitude of thebeating decreased

Furthermore the amplitudes of the theoreticalwaveformsin Figures 11 and 12 are different from the experimental resultin Figure 13 This may because af the following (a) The effectof mechanical damping with respect to stiffness was onlyconsidered and the mass damping was ignored in theoreticalanalysis which can also weaken the transient vibration faster(b) The first natural frequency of the cantilever system wascalculated and measured by theoretical and experimentalmethods as 815Hz and 812Hz respectively The little dif-ference made that the max amplitudes from theoretical andexperimental results did not appear at the same frequency(c) In theoretical analysis the tip-mass was equated by theinertia force and inertia moment acting at the free end ofthe cantilever so the total length for theoretical analysiswas just the length of the beam of 1204mm Howeverduring experiments the tip-displacement was measured at

12 Shock and Vibration

005

0

minus0050 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6

0 1 2 3 4 5 6 0 1 2 3 4 5 6

0 1 2 3 4 5 6 0 1 2 3 4 5 6

0 1 2 3 4 5 6

0

0

1 2 3 4 5 6

02

01

0

minus01

minus02

02

01

0

0

minus01

minus02

015

01

005

minus005

minus01

008

006

004

002

0

minus002

minus004

minus006

minus008

008

006

004

002

0

minus002

minus004

minus006

minus008

008

006

004

002

minus002

minus004

minus006

minus008

2

15

1

05

0

minus05

minus1

minus15

minus2

01

005

0

minus005

minus01Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at735

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

795

Hz (

mm

)

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at855

Hz (

mm

)

tran

sient

to st

eady

at755

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mTh

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

815

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

875

Hz (

mm

)

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at775

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

835

Hz (

mm

)

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at895

Hz (

mm

)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Figure 11 Theoretical tip displacement responses with the second-type mode shape of the beam from transient to steady state

the approximate position of 119909 = 1204mm This is justan approximate value because Eddy-Current Sensor cannotgive exact measurement results of the displacement at thejunction between the cantilever and the tip-mass This maycause that the amplitude level from theoretical discussionwashigher than from experimental measurement In additionthe amplitude of the instantaneous base displacement fromthe exciter may be not the fixed value 001mm at the startinginstant Therefore the maximum displacement responsesduring the transient state in the experimental measurementsmay shift away from the real experimental results

The theoretical responses with two improved types ofmode shapes and the experimental displacement responsesversus varying positions of beam and the frequencies atsteady state are shown in Figures 14ndash16 respectively Figures14 and 15 show two types of theoretical displacements atthe positions of 119909 = 364mm 574mm 784mm 994mmand 1204mmwith continuous varying frequencies In exper-iments the displacement responses were measured at thesame positions with theoretical calculations but with somecertain frequencies of 782Hz 792Hz 802Hz 807Hz81Hz 812Hz 817Hz 82Hz 83Hz and 84Hz as plottedin Figure 16 It is noticed that when the frequency of baseexcitation approached to the first natural frequency of thecantilever system the amplitude of forced vibration willrapidly increase and become maximum for the case when

the frequency of the excitation coincides with the naturalfrequency Table 3 lists the values of the first natural frequen-cies and the displacement responses from theoretical andexperimental results at some certain positions long the beamunder the first natural frequencies It can be seen that thetheoretical and experimental results are matched very close

Table 3 The first natural frequencies and the resonantdisplacements at some certain positions

In a word the theoretical results with two different modeshapes derived by two improved analytical methods haveextremely the same behaviors at transient and steady stateswith respect to varying frequencies from base excitationThedisplacement responses indicate close agreement betweenthe experimental and analytical results All of these resultsgive full verification for the improvement of the analyticalmethods in this paper utilized in recalculating the modeshapes of a cantilever beam when an offset distance existsbetween the gravity center of the tip-mass and the point ofattachment

5 Conclusions

This paper presents two improved analytical methods ofcalculation for natural frequencies and mode shapes of auniform cantilever beam carrying a tip-mass under an input

Shock and Vibration 13

005

0

minus005Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at735

Hz (

mm

)

0 1 2 3 4 5 6

Time (s)

0 1 2 3 4 5 6

0 1 2 3 4 5 6

02

01

0

minus01

minus02

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at795

Hz (

mm

)

0

015

01

005

minus005

minus01

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at855

Hz (

mm

)

0 1 2 3 4 5 6

Time (s)

008

006

004

002

0

minus002

minus004

minus006

minus008tran

sient

to st

eady

at755

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

m

2

15

1

05

0

minus05

minus1

minus15

minus2Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at815

Hz (

mm

)

0 1 2 3 4 5 6

008

006

004

002

0

minus002

minus004

minus006

minus008Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at875

Hz (

mm

)

0 1 2 3 4 5 6

01

005

0

minus005

minus01Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at775

Hz (

mm

)

0 1 2 3 4 5 6

Time (s)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

0 1 2 3 4 5 6

0 1 2 3 4 5 6

0

008

006

004

002

minus002

minus004

minus006

minus008Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at895

Hz (

mm

)

02

01

0

minus01

minus02

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at835

Hz (

mm

)Figure 12 Theoretical tip displacement responses with third-type mode shape of the beam from transient to steady state

Table 3 The first natural frequencies and the resonant displacements at some certain positions

Resonant frequency or positions The values of frequencies and displacement responses (Hz or mm) ErrorsTheory based on 119880(2)

1(119909) Theory based on 119880(3)

1(119909) Experiment

The 1st natural frequency 815 815 812 037119909 = 364mm 01285 01285 01277 062119909 = 574mm 03022 03022 03001 069119909 = 784mm 05279 05279 05246 063119909 = 994mm 07905 07905 07855 063119909 = 1204mm 10742 10742 10671 066

base excitation using forced vibration theory and the methodof separation of variables A simplified cantilever model byreplacing the tip-mass with an equivalent inertial force andinertial moment acting at the free end of the cantilever beamwas detailed In addition using the Dirac Delta Functionthe base transverse excitation the concentrated equivalentinertia force and inertia moment were represented in theform of distributed loads which can make the method ofseparation of variables be utilized properly In this case sometypical results of natural frequencies and mode shapes of thecantilever beam with a tip-mass were derived using the rein-forced method of separation of variables and the developedforced vibration respectively Furthermore based on modalanalysis method the transient and steady displacement

responses of the cantilever system with a tip-mass and a sinebase excitation were presented in which two different types ofmode shapes calculated by two improved analytical methodswere used Finally an experimental cantilever system wasfabricated and some practical displacement responses weremeasured

From the results of the typical natural frequencies andmode shapes we can easily found that the mode shapesfrom the first improved type were identical to the modeshapes from the second type whether the mode shapes fromthe lower order or from the higher order even the offsetdistance was large The comparing results of the transientand steady displacement responses from the theoreticalanalysis and experimental measurements further proved the

14 Shock and Vibration

005

0

minus0050 2 4 6 8 10 12

0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12

0 2 4 6 8 10 120 2 4 6 8 10 12

0 2 4 6 8 10 12 0 2 4 6 8 10 120 2 4 6 8 10 12

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at732

Hz (

mm

)

0

008

006

004

002

minus002

minus004

minus006

minus008

0

008

006

004

002

minus002

minus004

minus006

minus008

0

008

006

004

002

minus002

minus004

minus006

minus008

0

015

01

005

minus005

minus01

01

005

0

minus005

minus01

2

15

1

05

0

minus05

minus1

minus15

minus2

03

02

01

0

minus01

minus02

03

02

01

0

minus01

minus02

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at752

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at772

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at792

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at812

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at832

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at852

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at872

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at892

Hz (

mm

)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Figure 13 Experimental displacement responses at the position of 119909 = 1204mm from transient to steady state

12

1

08

06

04

02

0120 100 80 60 40 20

78

80

82

84Theo

retic

al st

eady

disp

lace

men

t (m

m)

Position at the beam (mm)Freq

uency

(Hz)

Figure 14 Theoretical steady-displacement responses versus posi-tions and frequencies with mode 119880(2)

1(119909)

improvements of the analytical methods for calculating themode shapes of the cantilever beam with a tip-mass anda base excitation In addition the amplitudes of the modeshapes decreased with the increasing mass value of the tip-mass when the moment of inertia and offset were constantsThe amplitude of the first-order mode shape increased but

12

1

08

06

04

02

0120 100 80 60 40 20

78

80

82

84Theo

retic

al st

eady

disp

lace

men

t (m

m)

Position at the beam (mm)Freq

uency

(Hz)

Figure 15 Theoretical steady-displacement responses versus posi-tions and frequencies with mode 119880(3)

1(119909)

the other nine mode shapes decreased with the increasingmoment of inertia when the mass and offset were set asconstants Conversely the amplitude of the first mode shapedecreased but the other ninemode shapes increased with theincreasing offset distance under the condition that the massand moment of inertia were constants Globally the effect

Shock and Vibration 15

12

1

08

06

04

02

0

Expe

rimen

tal s

tead

y di

spla

cem

ent (

mm

)

120 100 80 60 40 20Position at the beam (mm)

78

80

82

84Freq

uency

(Hz)

Figure 16 Experimental steady-displacement responses versuspositions and frequencies

from varying combinations of the mass and moment inertiaof the tip-mass and the offset distance for the lower-ordermode shapes was stronger than for the higher ones

Furthermore it is clearly noticed that the mass of tip-mass can reduce the values of the natural frequencies andthe increasing mass value gave the decreasing frequencieswhen the moment of inertia of the tip-mass and the offsetdistance were set as constants Similarly the moment ofinertia can further decrease the natural frequencies and thehigher moment of inertia resulted in the lower frequencieswhen the mass and the offset were constants Differently theoffset distance can reduce the first-order natural frequenciesbut increase the other nine order frequencies and the higheroffset distance led to the lower first-order natural frequenciesbut produced the higher other nine order frequencies whenthe mass and the moment of inertia were set to constant Inglobal view the influence from the varying mass momentof inertia and offset distance for the lower-order naturalfrequencies was larger than for the higher-order ones weatherthis influence was negative or positive

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the Natural Science Foundationsof China (no 51275380) and the Scientific and TechnologicalProject of Shaanxi Province (no 2012K06-36) The authorswould like to thank the reviewers for their valuable commentson this paper

References

[1] P A A Laura J L Pombo and E A Susemihl ldquoA note on thevibrations of a clamped-free beam with a mass at the free endrdquoJournal of Sound and Vibration vol 37 no 2 pp 161ndash168 1974

[2] B Rama Bhat andHWagner ldquoNatural frequencies of a uniformcantilever with a tip mass slender in the axial directionrdquo Journalof Sound and Vibration vol 45 no 2 pp 304ndash307 1976

[3] C W S To ldquoVibration of a cantilever beam with a baseexcitation and tip massrdquo Journal of Sound and Vibration vol83 no 4 pp 445ndash460 1982

[4] P A A Laura ldquoComments on lsquoVibration of a cantilever beamwith a base excitation and tip massrsquordquo Journal of Sound andVibration vol 88 no 4 p 569 1983

[5] C W S To ldquoDiscussion on lsquovibration of a cantilever beam witha base excitation and tip massrsquordquo Journal of Sound and Vibrationvol 97 no 2 pp 349ndash351 1984

[6] R G Jacquot ldquoFurther comments on lsquoVibration of a cantileverbeam with a base excitation and tip massrsquordquo Journal of Sound andVibration vol 93 no 2 pp 312ndash313 1984

[7] S AMousavi Lajimi andG R Heppler ldquoComments on lsquonaturalfrequencies of a uniform cantilever with a tip mass slender inthe axial directionrsquordquo Journal of Sound and Vibration vol 331 no12 pp 2964ndash2968 2012

[8] R B Bhat ldquoAuthors response to the comments on lsquonaturalfrequencies of a uniform cantilever with a tip mass slender inthe axial directionrsquordquo Journal of Sound and Vibration vol 331 no12 pp 2962ndash2963 2012

[9] E Esmailzadeh and G Nakhaie-Jazar ldquoPeriodic behavior ofa cantilever beam with end mass subjected to harmonic baseexcitationrdquo International Journal of Non-Linear Mechanics vol33 no 4 pp 567ndash577 1998

[10] H Abramovich and O Hamburger ldquoVibration of a cantilevertimoshenko beam with a tip massrdquo Journal of Sound andVibration vol 148 no 1 pp 162ndash170 1991

[11] B Pratiher ldquoVibration control of a transversely excited can-tilever beam with tip massrdquo Archive of Applied Mechanics vol82 no 1 pp 31ndash42 2012

[12] M Kim M Hoegen J Dugundji and B L Wardle ldquoModelingand experimental verification of proof mass effects on vibrationenergy harvester performancerdquo Smart Materials and Structuresvol 19 Article ID 045023 2010

[13] Y T Li and R Wong ldquoIntegral and series representations ofthe Dirac delta functionrdquo Communications on Pure and AppliedAnalysis vol 7 no 2 pp 229ndash247 2008

[14] M Al-Gwaiz Sturm-LiouvilleTheory and its Applications chap-ter 3 Springer London UK 2008

[15] F Bao-lian ldquoGeneralized reciprocal theorems and their appli-cationsrdquo Applied Mathematics and Mechanics vol 23 no 2 pp203ndash210 2001

[16] C Cai H Zheng M Khan and K HungModeling of MaterialDamping Properties in ANSYS Defense Systems Division Insti-tute of High Performance Computing 2002

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Shock and Vibration

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Civil EngineeringAdvances in

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Chemical EngineeringInternational Journal of Antennas and

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DistributedSensor Networks

International Journal of

8 Shock and Vibration

088087086085

0 05 1 0 05 1 0 05 1

0 05 1 0 05 1

0 05 10 05 1

0 05 1 0 05 1

0 05 1 0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0

minus50

minus100

minus150

6420

minus2

times104

840

minus4

times105

0minus1minus2minus3

times107

0302010

03

02

01

0

04

02

0

04

02

0

151

050

151

050

210

minus1

210

minus1

210

minus1

210

minus1

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

084906

085736

087731

minus1308377

minus9308045

minus16072843

63490847

565642 times 104

minus172997 times 104

minus329762092

230021 times 104

667852 times 105

946242416

minus149912 times 106

minus306922 times 107

minus995048 times 10minus10

007811

030669

511030 times 10minus11

021118

040600

minus114494 times 10minus10

160244

minus012466

688252 times 10minus11

minus002686

007708

183045 times 10minus11

minus140613

minus005561

000000

000000

000000

000000

000000

007811

030669

021118

040600

160244

minus012466

minus002686

007708

minus140613

minus005561

Ui(x)

xL

Ui(x)

xL

Ui(x)

xL

Notes at xL Notes at xL Notes at xLValues ValuesValuesFirst-type mode shapes U(1)i

(x) Second-type mode shapes U(2)i

(x) Third-type mode shapes U(3)i

(x)

Figure 7 Three types of the first five mode shapes with the combinations of 120601 = 4 119896119871 = 4 and 119889119871 = 2

the natural mode shapes should be derived In fact usingBetti theorem [15] the orthogonality properties with respectto mass and stiffness can be obtained respectively as follows

int

119871

0

119898119880(119896)

119903(119909)119880(119896)

119894(119909) 119889119909 +119872119880

(119896)

119903(119871)119880(119896)

119894(119871)

+ 119903119888119872119880(119896)

119894(119871)

119889119880(119896)

119903(119871)

119889119909

+ 119903119888119872119880(119896)

119903(119871)

119889119880(119896)

119894(119871)

119889119909

+ (119869 + 1199032

119888119872)

119889119880(119896)

119894(119871)

119889119909

119889119880(119896)

119903(119871)

119889119909

= (120575(119896)

119903119894)

2

(119896 = 1 2 119903 119894 = 1 2 )

(34)

int

119871

0

119864119868

1198892119880(119896)

119894(119909)

1198891199092

1198892119880(119896)

119903(119909)

1198891199092

119889119909 = (120596119903120575(119896)

119903119894)

2

(119896 = 1 2 119903 119894 = 1 2 )

(35)

In this case the normalized mode shapes can be defined as

119880(119896)

119903119873(119909) =

119880(119896)

119903(119909)

120575(119896)

119903119903

(36)

where

(120575(119896)

119903119903)

2

= int

119871

0

119898[119880(119896)

119903(119909)]

2

119889119909 +119872[119880(119896)

119903(119871)]

2

+ 2119903119888119872119880(119896)

119903(119871) [

119889

119889119909

119880(119896)

119903(119871)]

+ (119869 +1198721199032

119888) [

119889

119889119909

119880(119896)

119903(119871)]

2

(37)

41 Theoretical Responses of the Cantilever System with a BaseSine Excitation According to the method of modal analysisthe solution of (8) has the form of

119906119886(119909 119905) =

infin

sum

119903=1

119880(119896)

119903119873(119909) 119902119903(119905) (38)

So inserting (38) into (8) gives

119864119868

infin

sum

119903=1

1198894119880(119896)

119903119873(119909)

1198891199094

119902119903(119905) + 119898

infin

sum

119903=1

119880(119896)

119903119873(119909)

1198892119902119903(119905)

1198891199052

= [119872120575 (119909 minus 119871) + 119903119888119872

119889120575 (119909 minus 119871)

119889119909

minus 119898]

1198892119887 (119905)

1198891199052

(39)

Considering the orthogonality properties of equations (34)and (35) let (39) be multiplied by119880(119896)

119894119873(119909) and integrated over

Shock and Vibration 9

192

191

19

0 05 1 0 05 1

0 05 1 0 05 1

0 05 1

0 05 10 05 1

0 05 10 05 1

0 05 1 0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0

minus200

minus400

minus600

0minus5minus10

times104

1

0

minus1

times106

times107

0minus2minus4minus6

02

01

0

02

01

0

06

04

02

0

06

04

02

0

15

1

05

0

15

1

05

0

210

minus1

210

minus1

210

minus1

210

minus1

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

000000

050000

100000

i = 1

000000

050000

100000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

189913

190621

192051

minus56420429

minus28138349

4723398

134513 times 104

minus110859 times 105

minus260952 times 104

minus615688 times 104

496921 times 104

145091 times 106

168417 times 105

241806 times 106

minus633899 times 107

137503 times 10minus9

004964

019458

minus238241 times 10minus11

028530

054402

159443

minus019150

minus283947 times 10minus11

minus004212

012050

771494 times 10minus11

minus140605

minus008755

000000

000000

000000 000000

000000

000000

004964

019458

028530

054402

159443

minus019150

minus004212

012050

minus140605

minus008755

Ui(x)

xL

Ui(x)

xL

Ui(x)

xL

Notes at xL Notes at xL Notes at xLValues ValuesValues

minus387890 times 10minus11

First-type mode shapes U(1)i

(x) Second-type mode shapes U(2)i

(x) Third-type mode shapes U(3)i

(x)

Figure 8 Three types of the first five mode shapes with the combinations of 120601 = 4 119896119871 = 4 and 119889119871 = 4

the domain 0 lt 119909 lt 119871 to give a set of independent ordinarydifferential equations as

1198892119902119894(119905)

1198891199052

+ 1205962

119894119902119894(119905) = 119891

(119896)

119894(119905) (40)

Considering the properties of Dirac Delta Function

int

119871

0

119880(119896)

119894119873(119909) 120575 (119909 minus 119871) 119889119909 = 119880

(119896)

119894119873(119871)

int

119871

0

119880(119896)

119894119873(119909)

119889120575 (119909 minus 119871)

119889119909

119889119909 = minus

119889119880(119896)

119894119873(119871)

119889119909

(41)

the generalized forces associated with the generalized coordi-nates 119902

119894(119905) can be obtained as

119891(119896)

119894(119905)

= [119872119880(119896)

119894119873(119871) minus 119903

119888119872

119889119880(119896)

119894119873(119871)

119889119909

minus 119898int

119871

0

119880(119896)

119894119873(119909) 119889119909]

1198892119887 (119905)

1198891199052

(42)

Furthermore considering the mechanical damping asRayleigh damping and ignoring the part of mass dampingthe damping coefficient 120577 of the bimorph beam can be deter-mined by the modal damping ratio which can be obtained

by experiment [16] The independent ordinary differentialequations with damping then can be written as

1198892119902119894(119905)

1198891199052

+ 2120596119894120577

119889119902119894(119905)

119889119905

+ 1205962

119894119902119894(119905)

= [119872119880(119896)

119894119873(119871) minus 119903

119888119872

119889119880(119896)

119894119873(119871)

119889119909

minus 119898int

119871

0

119880(119896)

119894119873(119909) 119889119909]

1198892119887 (119905)

1198891199052

(43)

Now (43) can be solved using Laplace transform Applyinga sine excitation 119887(119905) = 119861 sin(120596119905) to the base of cantileverstructure under zero initial conditions the transfer functionof the independent differential equations for the cantileverbeam with tip-mass under base displacement excitation canbe written as

119866119894(119904)

=

119876119894(119904)

119861119894(119904)

=

(minus1205962)[119872119880

(119896)

119894119873(119871)minus119903

119888119872(119889119880

(119896)

119894119873(119871) 119889119909)minus119898int

119871

0119880(119896)

119894119873(119909) 119889119909]

1199042+ 2120596119894120577119904 + 120596

2

119894

(44)

10 Shock and Vibration

where 119876119894(119904) and 119861

119894(119904) are the Laplace transforms of 119902

119894(119905) and

119887(119905) respectively The inverse transformation of the transferfunction will be further obtained as

119892119894(119905)

=

(minus1205962)[119872119880

(119896)

119894119873(119871)minus119903

119888119872(119889119880

(119896)

119894119873(119871) 119889119909)minus119898int

119871

0119880(119896)

119894119873(119909) 119889119909]

120596119894radic1 minus 120577

2

times 119890minus120596119894120577119905 sin(120596

119894radic1 minus 120577

2119905)

(45)

In this case the solutions of (43) can be solved by convolutiontheorem as

119902119894(119905) = 119887 (119905) lowast 119892

119894(119905) = int

119905

0

119887 (120591) 119892119894(119905 minus 120591) 119889120591 (46)

Finally substituting the result of (46) into (38) and after somelengthy algebraic manipulation the absolute displacementresponse of the cantilever beam with tip-mass under basetransverse excitation can be obtained as

119906119886(119909 120596 119905) =

infin

sum

119894=1

119880(119896)

119894119873(119909)

119861 [119872119880(119896)

119894119873(119871) minus 119903

119888119872(119889119880

(119896)

119894119873(119871) 119889119909) minus 119898int

119871

0119880(119896)

119894119873(119909) 119889119909]

radic1205964+ 2 (2120577

2minus 1) 120596

21205962

119894+ 1205964

119894

times [

1205963

120596119894radic1 minus 120577

2

119890minus120596119894120577119905 sin(120596

119894radic1 minus 120577

2119905 + 1206011) + 1205962 sin (120596119905 + 120601

2)]

(47)

where 1206011and 120601

2are determined by tan(120601

1) = 2120577radic1 minus 120577

2

((120596120596119894)2+ 21205772minus 1) and tan(120601

2) = minus2120577(120596120596

119894)(1 minus (120596120596

119894)2)

It can be seen that the solution consists of two parts abouttimeThe first term represents free vibration which will decaywith time exponentially The second term depending on theexcitation represents the forced vibration of the system

42 Experimental Responses and Comparisons with the Theo-retical Results In order to explain the absolute displacementresponse of the cantilever structure under base excitationexperimentally a typical cantilever beam in which theelongated beam was inserted and fixed into the tip-mass tomake sure the gravity center of tip-mass was collinear withthe beam was fabricated (as shown in Figure 9) The chosengeometric and physical properties of the cantilever beamincluding the tip-mass are given in Table 2

The complete experimental devices are shown inFigure 10 The electromagnetic vibrating table type SW-TFAcontrolled by a programmable controller type SW-F-2TF-2was used to create a specific harmonic input excitationThe cantilever beam with a tip-mass was clamped onthe vibrating top platform Eddy-Current Sensors typeDM6008 and ZA210800 were used to measure the inputdisplacement from base vibrating exciter and the absolutedynamic displacement from the beam respectively Allmeasurements from cantilever beam and vibrating exciterwere connected to Digital Oscilloscope type DS1204Band the Industrial Computer including a PC-based DataAcquisitionModule from ADLINK company type DAQ2213The Digital Oscilloscope and Industrial Computer displayedand processed the measurement results using the DAQPilotsoftware and the results from subsequent analysis using theMATLAB software

Table 2 Characteristic properties of the cantilever beam includingtip-mass

Physical properties Beam Tip-massYoungrsquos modulus 119864 (GPa) 69 mdashDensity 120588

119904or 120588119898(kgm3) 2630 7753

Length 119871 or 119871119898(mm) 1204 156

Thickness 119905119904or 119905119898(mm) 49 295

Width 119908 (mm) 102 102Total mass119872 (g)a mdash 3383Offset 119903

119888(mm)b mdash 801

Rotary inertia 119869 (kgsdotm2)b c mdash 333 times 10minus6aThe total mass of the tip-mass including the elongated part of beam wascalculated by119872 = 119871

119898119905119898119908120588119898+ 119871119904119905119904119908(120588119904minus 120588119898) where 119871

119904= 10mm

bThe offset distance and the rotary inertia were calculated according to thegeometry and material properties of the beam and tip-mass using equations119903119888= (120588

1198981199051198981198712

119898minus 1205881198981199051199041198712

119904+ 1205881199041199051199041198712

119904)(2[120588

119898119905119898119871119898+ 119871119904119905119904(120588119904minus 120588119898)])

119869 = 120588119898119871119898119905119898119908 [(112)(119871

2

119898+ 1199052

119898) + ((12)119871

119898minus 119903119888)2] + 119871

119904119905119904119908(120588119904minus

120588119898)[(112)(119871

2

119904+ 1199052

119904) + (119903119888minus (12)119871

119904)2]

cThe corresponding parameters 120601 119896119871 and 119903119888119871 were calculated as follows

120601 = 119872(120588119904119905119904119908) = 21377 119896119871 = radic119869119872119871 = 00824 and 119903

119888119871 = 00665

The analytical model defined by (47) in which two typesof mode shapes derived by two improved analytical methodsare utilized can be used to analyze the steady and transientdisplacement responses of the cantilever beam with a tip-mass and an input base excitation based on the chosenvarying frequencies time and positions along the lengthof the beam Typically the displacement responses frompositions along 119909-axis at 119909

1= 364mm 119909

2= 574mm

1199093

= 784mm 1199094

= 994mm and 1199095

= 1204mmoff the origin point (as shown in Figure 9) were measured

Shock and Vibration 11

y

b(t)

x1

x2

x3

x4

x5

Ls

L

t s

t m x

Lm

rc

Cg

o998400

Figure 9 Typical cantilever beam with a tip-mass

1 2

3

56

7

8

4

Tip-massCantilever beam

(7) Cantilever with a tip-mass

(1) Industrial computer including data acquisitionmodule type DAQ2213

(2) Digital oscilloscope type DS1204B(3) Programmable controller type SW-F-FTF-2(4) Power supply type DP1308A(5) Driver modules for Eddy-current sensors(6) Electromagnetic vibrating table type SW-TFA

(8) Eddy-current sensors type DM6008 and ZA210800

Figure 10 Complete experimental setup

by Eddy-Current Sensors The amplitude of the input sine-displacement excitation for analysis was measured as 119861 =

001mm and the damping ratio for the transverse vibrationaround the fundamental resonant frequency was found to be120577 = 00025 by Logarithmic Decrement Method

In the following paragraphs we focus on the transientand steady responses of the cantilever system with a sine baseexcitation

Figures 11 and 12 show the theoretical displacementresponses at the free end of the beam from transient tosteady state with two improved mode shapes above at thefirst resonant frequency It can be seen obviously that thedisplacement responses with two improved mode shapeswe proposed are almost identical This richly proves thatthe first-type mode shapes reformulated by the reinforcedmethod of separation of variables are equal to the second-type mode shapes derived by the developed method offorced vibration In addition it can be further obtainedthat when the frequency from the base excitation is veryclose but not exactly equal to the resonant frequency of freevibrations of the cantilever system a kind of vibration calledbeating occurs and the period of the beating increases as120596 approached 120596

1 This kind of beating decays exponentially

along the time due to the mechanical dampingFigure 13 shows the experimental displacement responses

at the position of 119909 = 1204mm from transient to steady

state As the same to the theoretical results experimental dis-placement response can be regarded as the sum of waveformsfrom transient and steady states But the transient responsevanished gradually with the time due to the mechanicaldamping and only the steady response left In addition thephenomenon of beating still existed The amplitude of thisbeating phenomenon reduced gradually and the closer theexciting frequency approached to the first natural frequencyof the cantilever system the faster the amplitude of thebeating decreased

Furthermore the amplitudes of the theoreticalwaveformsin Figures 11 and 12 are different from the experimental resultin Figure 13 This may because af the following (a) The effectof mechanical damping with respect to stiffness was onlyconsidered and the mass damping was ignored in theoreticalanalysis which can also weaken the transient vibration faster(b) The first natural frequency of the cantilever system wascalculated and measured by theoretical and experimentalmethods as 815Hz and 812Hz respectively The little dif-ference made that the max amplitudes from theoretical andexperimental results did not appear at the same frequency(c) In theoretical analysis the tip-mass was equated by theinertia force and inertia moment acting at the free end ofthe cantilever so the total length for theoretical analysiswas just the length of the beam of 1204mm Howeverduring experiments the tip-displacement was measured at

12 Shock and Vibration

005

0

minus0050 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6

0 1 2 3 4 5 6 0 1 2 3 4 5 6

0 1 2 3 4 5 6 0 1 2 3 4 5 6

0 1 2 3 4 5 6

0

0

1 2 3 4 5 6

02

01

0

minus01

minus02

02

01

0

0

minus01

minus02

015

01

005

minus005

minus01

008

006

004

002

0

minus002

minus004

minus006

minus008

008

006

004

002

0

minus002

minus004

minus006

minus008

008

006

004

002

minus002

minus004

minus006

minus008

2

15

1

05

0

minus05

minus1

minus15

minus2

01

005

0

minus005

minus01Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at735

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

795

Hz (

mm

)

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at855

Hz (

mm

)

tran

sient

to st

eady

at755

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mTh

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

815

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

875

Hz (

mm

)

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at775

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

835

Hz (

mm

)

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at895

Hz (

mm

)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Figure 11 Theoretical tip displacement responses with the second-type mode shape of the beam from transient to steady state

the approximate position of 119909 = 1204mm This is justan approximate value because Eddy-Current Sensor cannotgive exact measurement results of the displacement at thejunction between the cantilever and the tip-mass This maycause that the amplitude level from theoretical discussionwashigher than from experimental measurement In additionthe amplitude of the instantaneous base displacement fromthe exciter may be not the fixed value 001mm at the startinginstant Therefore the maximum displacement responsesduring the transient state in the experimental measurementsmay shift away from the real experimental results

The theoretical responses with two improved types ofmode shapes and the experimental displacement responsesversus varying positions of beam and the frequencies atsteady state are shown in Figures 14ndash16 respectively Figures14 and 15 show two types of theoretical displacements atthe positions of 119909 = 364mm 574mm 784mm 994mmand 1204mmwith continuous varying frequencies In exper-iments the displacement responses were measured at thesame positions with theoretical calculations but with somecertain frequencies of 782Hz 792Hz 802Hz 807Hz81Hz 812Hz 817Hz 82Hz 83Hz and 84Hz as plottedin Figure 16 It is noticed that when the frequency of baseexcitation approached to the first natural frequency of thecantilever system the amplitude of forced vibration willrapidly increase and become maximum for the case when

the frequency of the excitation coincides with the naturalfrequency Table 3 lists the values of the first natural frequen-cies and the displacement responses from theoretical andexperimental results at some certain positions long the beamunder the first natural frequencies It can be seen that thetheoretical and experimental results are matched very close

Table 3 The first natural frequencies and the resonantdisplacements at some certain positions

In a word the theoretical results with two different modeshapes derived by two improved analytical methods haveextremely the same behaviors at transient and steady stateswith respect to varying frequencies from base excitationThedisplacement responses indicate close agreement betweenthe experimental and analytical results All of these resultsgive full verification for the improvement of the analyticalmethods in this paper utilized in recalculating the modeshapes of a cantilever beam when an offset distance existsbetween the gravity center of the tip-mass and the point ofattachment

5 Conclusions

This paper presents two improved analytical methods ofcalculation for natural frequencies and mode shapes of auniform cantilever beam carrying a tip-mass under an input

Shock and Vibration 13

005

0

minus005Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at735

Hz (

mm

)

0 1 2 3 4 5 6

Time (s)

0 1 2 3 4 5 6

0 1 2 3 4 5 6

02

01

0

minus01

minus02

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at795

Hz (

mm

)

0

015

01

005

minus005

minus01

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at855

Hz (

mm

)

0 1 2 3 4 5 6

Time (s)

008

006

004

002

0

minus002

minus004

minus006

minus008tran

sient

to st

eady

at755

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

m

2

15

1

05

0

minus05

minus1

minus15

minus2Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at815

Hz (

mm

)

0 1 2 3 4 5 6

008

006

004

002

0

minus002

minus004

minus006

minus008Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at875

Hz (

mm

)

0 1 2 3 4 5 6

01

005

0

minus005

minus01Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at775

Hz (

mm

)

0 1 2 3 4 5 6

Time (s)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

0 1 2 3 4 5 6

0 1 2 3 4 5 6

0

008

006

004

002

minus002

minus004

minus006

minus008Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at895

Hz (

mm

)

02

01

0

minus01

minus02

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at835

Hz (

mm

)Figure 12 Theoretical tip displacement responses with third-type mode shape of the beam from transient to steady state

Table 3 The first natural frequencies and the resonant displacements at some certain positions

Resonant frequency or positions The values of frequencies and displacement responses (Hz or mm) ErrorsTheory based on 119880(2)

1(119909) Theory based on 119880(3)

1(119909) Experiment

The 1st natural frequency 815 815 812 037119909 = 364mm 01285 01285 01277 062119909 = 574mm 03022 03022 03001 069119909 = 784mm 05279 05279 05246 063119909 = 994mm 07905 07905 07855 063119909 = 1204mm 10742 10742 10671 066

base excitation using forced vibration theory and the methodof separation of variables A simplified cantilever model byreplacing the tip-mass with an equivalent inertial force andinertial moment acting at the free end of the cantilever beamwas detailed In addition using the Dirac Delta Functionthe base transverse excitation the concentrated equivalentinertia force and inertia moment were represented in theform of distributed loads which can make the method ofseparation of variables be utilized properly In this case sometypical results of natural frequencies and mode shapes of thecantilever beam with a tip-mass were derived using the rein-forced method of separation of variables and the developedforced vibration respectively Furthermore based on modalanalysis method the transient and steady displacement

responses of the cantilever system with a tip-mass and a sinebase excitation were presented in which two different types ofmode shapes calculated by two improved analytical methodswere used Finally an experimental cantilever system wasfabricated and some practical displacement responses weremeasured

From the results of the typical natural frequencies andmode shapes we can easily found that the mode shapesfrom the first improved type were identical to the modeshapes from the second type whether the mode shapes fromthe lower order or from the higher order even the offsetdistance was large The comparing results of the transientand steady displacement responses from the theoreticalanalysis and experimental measurements further proved the

14 Shock and Vibration

005

0

minus0050 2 4 6 8 10 12

0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12

0 2 4 6 8 10 120 2 4 6 8 10 12

0 2 4 6 8 10 12 0 2 4 6 8 10 120 2 4 6 8 10 12

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at732

Hz (

mm

)

0

008

006

004

002

minus002

minus004

minus006

minus008

0

008

006

004

002

minus002

minus004

minus006

minus008

0

008

006

004

002

minus002

minus004

minus006

minus008

0

015

01

005

minus005

minus01

01

005

0

minus005

minus01

2

15

1

05

0

minus05

minus1

minus15

minus2

03

02

01

0

minus01

minus02

03

02

01

0

minus01

minus02

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at752

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at772

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at792

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at812

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at832

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at852

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at872

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at892

Hz (

mm

)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Figure 13 Experimental displacement responses at the position of 119909 = 1204mm from transient to steady state

12

1

08

06

04

02

0120 100 80 60 40 20

78

80

82

84Theo

retic

al st

eady

disp

lace

men

t (m

m)

Position at the beam (mm)Freq

uency

(Hz)

Figure 14 Theoretical steady-displacement responses versus posi-tions and frequencies with mode 119880(2)

1(119909)

improvements of the analytical methods for calculating themode shapes of the cantilever beam with a tip-mass anda base excitation In addition the amplitudes of the modeshapes decreased with the increasing mass value of the tip-mass when the moment of inertia and offset were constantsThe amplitude of the first-order mode shape increased but

12

1

08

06

04

02

0120 100 80 60 40 20

78

80

82

84Theo

retic

al st

eady

disp

lace

men

t (m

m)

Position at the beam (mm)Freq

uency

(Hz)

Figure 15 Theoretical steady-displacement responses versus posi-tions and frequencies with mode 119880(3)

1(119909)

the other nine mode shapes decreased with the increasingmoment of inertia when the mass and offset were set asconstants Conversely the amplitude of the first mode shapedecreased but the other ninemode shapes increased with theincreasing offset distance under the condition that the massand moment of inertia were constants Globally the effect

Shock and Vibration 15

12

1

08

06

04

02

0

Expe

rimen

tal s

tead

y di

spla

cem

ent (

mm

)

120 100 80 60 40 20Position at the beam (mm)

78

80

82

84Freq

uency

(Hz)

Figure 16 Experimental steady-displacement responses versuspositions and frequencies

from varying combinations of the mass and moment inertiaof the tip-mass and the offset distance for the lower-ordermode shapes was stronger than for the higher ones

Furthermore it is clearly noticed that the mass of tip-mass can reduce the values of the natural frequencies andthe increasing mass value gave the decreasing frequencieswhen the moment of inertia of the tip-mass and the offsetdistance were set as constants Similarly the moment ofinertia can further decrease the natural frequencies and thehigher moment of inertia resulted in the lower frequencieswhen the mass and the offset were constants Differently theoffset distance can reduce the first-order natural frequenciesbut increase the other nine order frequencies and the higheroffset distance led to the lower first-order natural frequenciesbut produced the higher other nine order frequencies whenthe mass and the moment of inertia were set to constant Inglobal view the influence from the varying mass momentof inertia and offset distance for the lower-order naturalfrequencies was larger than for the higher-order ones weatherthis influence was negative or positive

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the Natural Science Foundationsof China (no 51275380) and the Scientific and TechnologicalProject of Shaanxi Province (no 2012K06-36) The authorswould like to thank the reviewers for their valuable commentson this paper

References

[1] P A A Laura J L Pombo and E A Susemihl ldquoA note on thevibrations of a clamped-free beam with a mass at the free endrdquoJournal of Sound and Vibration vol 37 no 2 pp 161ndash168 1974

[2] B Rama Bhat andHWagner ldquoNatural frequencies of a uniformcantilever with a tip mass slender in the axial directionrdquo Journalof Sound and Vibration vol 45 no 2 pp 304ndash307 1976

[3] C W S To ldquoVibration of a cantilever beam with a baseexcitation and tip massrdquo Journal of Sound and Vibration vol83 no 4 pp 445ndash460 1982

[4] P A A Laura ldquoComments on lsquoVibration of a cantilever beamwith a base excitation and tip massrsquordquo Journal of Sound andVibration vol 88 no 4 p 569 1983

[5] C W S To ldquoDiscussion on lsquovibration of a cantilever beam witha base excitation and tip massrsquordquo Journal of Sound and Vibrationvol 97 no 2 pp 349ndash351 1984

[6] R G Jacquot ldquoFurther comments on lsquoVibration of a cantileverbeam with a base excitation and tip massrsquordquo Journal of Sound andVibration vol 93 no 2 pp 312ndash313 1984

[7] S AMousavi Lajimi andG R Heppler ldquoComments on lsquonaturalfrequencies of a uniform cantilever with a tip mass slender inthe axial directionrsquordquo Journal of Sound and Vibration vol 331 no12 pp 2964ndash2968 2012

[8] R B Bhat ldquoAuthors response to the comments on lsquonaturalfrequencies of a uniform cantilever with a tip mass slender inthe axial directionrsquordquo Journal of Sound and Vibration vol 331 no12 pp 2962ndash2963 2012

[9] E Esmailzadeh and G Nakhaie-Jazar ldquoPeriodic behavior ofa cantilever beam with end mass subjected to harmonic baseexcitationrdquo International Journal of Non-Linear Mechanics vol33 no 4 pp 567ndash577 1998

[10] H Abramovich and O Hamburger ldquoVibration of a cantilevertimoshenko beam with a tip massrdquo Journal of Sound andVibration vol 148 no 1 pp 162ndash170 1991

[11] B Pratiher ldquoVibration control of a transversely excited can-tilever beam with tip massrdquo Archive of Applied Mechanics vol82 no 1 pp 31ndash42 2012

[12] M Kim M Hoegen J Dugundji and B L Wardle ldquoModelingand experimental verification of proof mass effects on vibrationenergy harvester performancerdquo Smart Materials and Structuresvol 19 Article ID 045023 2010

[13] Y T Li and R Wong ldquoIntegral and series representations ofthe Dirac delta functionrdquo Communications on Pure and AppliedAnalysis vol 7 no 2 pp 229ndash247 2008

[14] M Al-Gwaiz Sturm-LiouvilleTheory and its Applications chap-ter 3 Springer London UK 2008

[15] F Bao-lian ldquoGeneralized reciprocal theorems and their appli-cationsrdquo Applied Mathematics and Mechanics vol 23 no 2 pp203ndash210 2001

[16] C Cai H Zheng M Khan and K HungModeling of MaterialDamping Properties in ANSYS Defense Systems Division Insti-tute of High Performance Computing 2002

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Shock and Vibration

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International Journal of

Shock and Vibration 9

192

191

19

0 05 1 0 05 1

0 05 1 0 05 1

0 05 1

0 05 10 05 1

0 05 10 05 1

0 05 1 0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0 05 1

0

minus200

minus400

minus600

0minus5minus10

times104

1

0

minus1

times106

times107

0minus2minus4minus6

02

01

0

02

01

0

06

04

02

0

06

04

02

0

15

1

05

0

15

1

05

0

210

minus1

210

minus1

210

minus1

210

minus1

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

000000

050000

100000

i = 1

000000

050000

100000

000000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

000000

050000

100000

i = 1

000000

050000

100000

050000

100000

000000

050000

100000

000000

050000

100000

i = 2

i = 3

i = 4

i = 5

189913

190621

192051

minus56420429

minus28138349

4723398

134513 times 104

minus110859 times 105

minus260952 times 104

minus615688 times 104

496921 times 104

145091 times 106

168417 times 105

241806 times 106

minus633899 times 107

137503 times 10minus9

004964

019458

minus238241 times 10minus11

028530

054402

159443

minus019150

minus283947 times 10minus11

minus004212

012050

771494 times 10minus11

minus140605

minus008755

000000

000000

000000 000000

000000

000000

004964

019458

028530

054402

159443

minus019150

minus004212

012050

minus140605

minus008755

Ui(x)

xL

Ui(x)

xL

Ui(x)

xL

Notes at xL Notes at xL Notes at xLValues ValuesValues

minus387890 times 10minus11

First-type mode shapes U(1)i

(x) Second-type mode shapes U(2)i

(x) Third-type mode shapes U(3)i

(x)

Figure 8 Three types of the first five mode shapes with the combinations of 120601 = 4 119896119871 = 4 and 119889119871 = 4

the domain 0 lt 119909 lt 119871 to give a set of independent ordinarydifferential equations as

1198892119902119894(119905)

1198891199052

+ 1205962

119894119902119894(119905) = 119891

(119896)

119894(119905) (40)

Considering the properties of Dirac Delta Function

int

119871

0

119880(119896)

119894119873(119909) 120575 (119909 minus 119871) 119889119909 = 119880

(119896)

119894119873(119871)

int

119871

0

119880(119896)

119894119873(119909)

119889120575 (119909 minus 119871)

119889119909

119889119909 = minus

119889119880(119896)

119894119873(119871)

119889119909

(41)

the generalized forces associated with the generalized coordi-nates 119902

119894(119905) can be obtained as

119891(119896)

119894(119905)

= [119872119880(119896)

119894119873(119871) minus 119903

119888119872

119889119880(119896)

119894119873(119871)

119889119909

minus 119898int

119871

0

119880(119896)

119894119873(119909) 119889119909]

1198892119887 (119905)

1198891199052

(42)

Furthermore considering the mechanical damping asRayleigh damping and ignoring the part of mass dampingthe damping coefficient 120577 of the bimorph beam can be deter-mined by the modal damping ratio which can be obtained

by experiment [16] The independent ordinary differentialequations with damping then can be written as

1198892119902119894(119905)

1198891199052

+ 2120596119894120577

119889119902119894(119905)

119889119905

+ 1205962

119894119902119894(119905)

= [119872119880(119896)

119894119873(119871) minus 119903

119888119872

119889119880(119896)

119894119873(119871)

119889119909

minus 119898int

119871

0

119880(119896)

119894119873(119909) 119889119909]

1198892119887 (119905)

1198891199052

(43)

Now (43) can be solved using Laplace transform Applyinga sine excitation 119887(119905) = 119861 sin(120596119905) to the base of cantileverstructure under zero initial conditions the transfer functionof the independent differential equations for the cantileverbeam with tip-mass under base displacement excitation canbe written as

119866119894(119904)

=

119876119894(119904)

119861119894(119904)

=

(minus1205962)[119872119880

(119896)

119894119873(119871)minus119903

119888119872(119889119880

(119896)

119894119873(119871) 119889119909)minus119898int

119871

0119880(119896)

119894119873(119909) 119889119909]

1199042+ 2120596119894120577119904 + 120596

2

119894

(44)

10 Shock and Vibration

where 119876119894(119904) and 119861

119894(119904) are the Laplace transforms of 119902

119894(119905) and

119887(119905) respectively The inverse transformation of the transferfunction will be further obtained as

119892119894(119905)

=

(minus1205962)[119872119880

(119896)

119894119873(119871)minus119903

119888119872(119889119880

(119896)

119894119873(119871) 119889119909)minus119898int

119871

0119880(119896)

119894119873(119909) 119889119909]

120596119894radic1 minus 120577

2

times 119890minus120596119894120577119905 sin(120596

119894radic1 minus 120577

2119905)

(45)

In this case the solutions of (43) can be solved by convolutiontheorem as

119902119894(119905) = 119887 (119905) lowast 119892

119894(119905) = int

119905

0

119887 (120591) 119892119894(119905 minus 120591) 119889120591 (46)

Finally substituting the result of (46) into (38) and after somelengthy algebraic manipulation the absolute displacementresponse of the cantilever beam with tip-mass under basetransverse excitation can be obtained as

119906119886(119909 120596 119905) =

infin

sum

119894=1

119880(119896)

119894119873(119909)

119861 [119872119880(119896)

119894119873(119871) minus 119903

119888119872(119889119880

(119896)

119894119873(119871) 119889119909) minus 119898int

119871

0119880(119896)

119894119873(119909) 119889119909]

radic1205964+ 2 (2120577

2minus 1) 120596

21205962

119894+ 1205964

119894

times [

1205963

120596119894radic1 minus 120577

2

119890minus120596119894120577119905 sin(120596

119894radic1 minus 120577

2119905 + 1206011) + 1205962 sin (120596119905 + 120601

2)]

(47)

where 1206011and 120601

2are determined by tan(120601

1) = 2120577radic1 minus 120577

2

((120596120596119894)2+ 21205772minus 1) and tan(120601

2) = minus2120577(120596120596

119894)(1 minus (120596120596

119894)2)

It can be seen that the solution consists of two parts abouttimeThe first term represents free vibration which will decaywith time exponentially The second term depending on theexcitation represents the forced vibration of the system

42 Experimental Responses and Comparisons with the Theo-retical Results In order to explain the absolute displacementresponse of the cantilever structure under base excitationexperimentally a typical cantilever beam in which theelongated beam was inserted and fixed into the tip-mass tomake sure the gravity center of tip-mass was collinear withthe beam was fabricated (as shown in Figure 9) The chosengeometric and physical properties of the cantilever beamincluding the tip-mass are given in Table 2

The complete experimental devices are shown inFigure 10 The electromagnetic vibrating table type SW-TFAcontrolled by a programmable controller type SW-F-2TF-2was used to create a specific harmonic input excitationThe cantilever beam with a tip-mass was clamped onthe vibrating top platform Eddy-Current Sensors typeDM6008 and ZA210800 were used to measure the inputdisplacement from base vibrating exciter and the absolutedynamic displacement from the beam respectively Allmeasurements from cantilever beam and vibrating exciterwere connected to Digital Oscilloscope type DS1204Band the Industrial Computer including a PC-based DataAcquisitionModule from ADLINK company type DAQ2213The Digital Oscilloscope and Industrial Computer displayedand processed the measurement results using the DAQPilotsoftware and the results from subsequent analysis using theMATLAB software

Table 2 Characteristic properties of the cantilever beam includingtip-mass

Physical properties Beam Tip-massYoungrsquos modulus 119864 (GPa) 69 mdashDensity 120588

119904or 120588119898(kgm3) 2630 7753

Length 119871 or 119871119898(mm) 1204 156

Thickness 119905119904or 119905119898(mm) 49 295

Width 119908 (mm) 102 102Total mass119872 (g)a mdash 3383Offset 119903

119888(mm)b mdash 801

Rotary inertia 119869 (kgsdotm2)b c mdash 333 times 10minus6aThe total mass of the tip-mass including the elongated part of beam wascalculated by119872 = 119871

119898119905119898119908120588119898+ 119871119904119905119904119908(120588119904minus 120588119898) where 119871

119904= 10mm

bThe offset distance and the rotary inertia were calculated according to thegeometry and material properties of the beam and tip-mass using equations119903119888= (120588

1198981199051198981198712

119898minus 1205881198981199051199041198712

119904+ 1205881199041199051199041198712

119904)(2[120588

119898119905119898119871119898+ 119871119904119905119904(120588119904minus 120588119898)])

119869 = 120588119898119871119898119905119898119908 [(112)(119871

2

119898+ 1199052

119898) + ((12)119871

119898minus 119903119888)2] + 119871

119904119905119904119908(120588119904minus

120588119898)[(112)(119871

2

119904+ 1199052

119904) + (119903119888minus (12)119871

119904)2]

cThe corresponding parameters 120601 119896119871 and 119903119888119871 were calculated as follows

120601 = 119872(120588119904119905119904119908) = 21377 119896119871 = radic119869119872119871 = 00824 and 119903

119888119871 = 00665

The analytical model defined by (47) in which two typesof mode shapes derived by two improved analytical methodsare utilized can be used to analyze the steady and transientdisplacement responses of the cantilever beam with a tip-mass and an input base excitation based on the chosenvarying frequencies time and positions along the lengthof the beam Typically the displacement responses frompositions along 119909-axis at 119909

1= 364mm 119909

2= 574mm

1199093

= 784mm 1199094

= 994mm and 1199095

= 1204mmoff the origin point (as shown in Figure 9) were measured

Shock and Vibration 11

y

b(t)

x1

x2

x3

x4

x5

Ls

L

t s

t m x

Lm

rc

Cg

o998400

Figure 9 Typical cantilever beam with a tip-mass

1 2

3

56

7

8

4

Tip-massCantilever beam

(7) Cantilever with a tip-mass

(1) Industrial computer including data acquisitionmodule type DAQ2213

(2) Digital oscilloscope type DS1204B(3) Programmable controller type SW-F-FTF-2(4) Power supply type DP1308A(5) Driver modules for Eddy-current sensors(6) Electromagnetic vibrating table type SW-TFA

(8) Eddy-current sensors type DM6008 and ZA210800

Figure 10 Complete experimental setup

by Eddy-Current Sensors The amplitude of the input sine-displacement excitation for analysis was measured as 119861 =

001mm and the damping ratio for the transverse vibrationaround the fundamental resonant frequency was found to be120577 = 00025 by Logarithmic Decrement Method

In the following paragraphs we focus on the transientand steady responses of the cantilever system with a sine baseexcitation

Figures 11 and 12 show the theoretical displacementresponses at the free end of the beam from transient tosteady state with two improved mode shapes above at thefirst resonant frequency It can be seen obviously that thedisplacement responses with two improved mode shapeswe proposed are almost identical This richly proves thatthe first-type mode shapes reformulated by the reinforcedmethod of separation of variables are equal to the second-type mode shapes derived by the developed method offorced vibration In addition it can be further obtainedthat when the frequency from the base excitation is veryclose but not exactly equal to the resonant frequency of freevibrations of the cantilever system a kind of vibration calledbeating occurs and the period of the beating increases as120596 approached 120596

1 This kind of beating decays exponentially

along the time due to the mechanical dampingFigure 13 shows the experimental displacement responses

at the position of 119909 = 1204mm from transient to steady

state As the same to the theoretical results experimental dis-placement response can be regarded as the sum of waveformsfrom transient and steady states But the transient responsevanished gradually with the time due to the mechanicaldamping and only the steady response left In addition thephenomenon of beating still existed The amplitude of thisbeating phenomenon reduced gradually and the closer theexciting frequency approached to the first natural frequencyof the cantilever system the faster the amplitude of thebeating decreased

Furthermore the amplitudes of the theoreticalwaveformsin Figures 11 and 12 are different from the experimental resultin Figure 13 This may because af the following (a) The effectof mechanical damping with respect to stiffness was onlyconsidered and the mass damping was ignored in theoreticalanalysis which can also weaken the transient vibration faster(b) The first natural frequency of the cantilever system wascalculated and measured by theoretical and experimentalmethods as 815Hz and 812Hz respectively The little dif-ference made that the max amplitudes from theoretical andexperimental results did not appear at the same frequency(c) In theoretical analysis the tip-mass was equated by theinertia force and inertia moment acting at the free end ofthe cantilever so the total length for theoretical analysiswas just the length of the beam of 1204mm Howeverduring experiments the tip-displacement was measured at

12 Shock and Vibration

005

0

minus0050 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6

0 1 2 3 4 5 6 0 1 2 3 4 5 6

0 1 2 3 4 5 6 0 1 2 3 4 5 6

0 1 2 3 4 5 6

0

0

1 2 3 4 5 6

02

01

0

minus01

minus02

02

01

0

0

minus01

minus02

015

01

005

minus005

minus01

008

006

004

002

0

minus002

minus004

minus006

minus008

008

006

004

002

0

minus002

minus004

minus006

minus008

008

006

004

002

minus002

minus004

minus006

minus008

2

15

1

05

0

minus05

minus1

minus15

minus2

01

005

0

minus005

minus01Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at735

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

795

Hz (

mm

)

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at855

Hz (

mm

)

tran

sient

to st

eady

at755

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mTh

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

815

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

875

Hz (

mm

)

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at775

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

835

Hz (

mm

)

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at895

Hz (

mm

)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Figure 11 Theoretical tip displacement responses with the second-type mode shape of the beam from transient to steady state

the approximate position of 119909 = 1204mm This is justan approximate value because Eddy-Current Sensor cannotgive exact measurement results of the displacement at thejunction between the cantilever and the tip-mass This maycause that the amplitude level from theoretical discussionwashigher than from experimental measurement In additionthe amplitude of the instantaneous base displacement fromthe exciter may be not the fixed value 001mm at the startinginstant Therefore the maximum displacement responsesduring the transient state in the experimental measurementsmay shift away from the real experimental results

The theoretical responses with two improved types ofmode shapes and the experimental displacement responsesversus varying positions of beam and the frequencies atsteady state are shown in Figures 14ndash16 respectively Figures14 and 15 show two types of theoretical displacements atthe positions of 119909 = 364mm 574mm 784mm 994mmand 1204mmwith continuous varying frequencies In exper-iments the displacement responses were measured at thesame positions with theoretical calculations but with somecertain frequencies of 782Hz 792Hz 802Hz 807Hz81Hz 812Hz 817Hz 82Hz 83Hz and 84Hz as plottedin Figure 16 It is noticed that when the frequency of baseexcitation approached to the first natural frequency of thecantilever system the amplitude of forced vibration willrapidly increase and become maximum for the case when

the frequency of the excitation coincides with the naturalfrequency Table 3 lists the values of the first natural frequen-cies and the displacement responses from theoretical andexperimental results at some certain positions long the beamunder the first natural frequencies It can be seen that thetheoretical and experimental results are matched very close

Table 3 The first natural frequencies and the resonantdisplacements at some certain positions

In a word the theoretical results with two different modeshapes derived by two improved analytical methods haveextremely the same behaviors at transient and steady stateswith respect to varying frequencies from base excitationThedisplacement responses indicate close agreement betweenthe experimental and analytical results All of these resultsgive full verification for the improvement of the analyticalmethods in this paper utilized in recalculating the modeshapes of a cantilever beam when an offset distance existsbetween the gravity center of the tip-mass and the point ofattachment

5 Conclusions

This paper presents two improved analytical methods ofcalculation for natural frequencies and mode shapes of auniform cantilever beam carrying a tip-mass under an input

Shock and Vibration 13

005

0

minus005Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at735

Hz (

mm

)

0 1 2 3 4 5 6

Time (s)

0 1 2 3 4 5 6

0 1 2 3 4 5 6

02

01

0

minus01

minus02

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at795

Hz (

mm

)

0

015

01

005

minus005

minus01

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at855

Hz (

mm

)

0 1 2 3 4 5 6

Time (s)

008

006

004

002

0

minus002

minus004

minus006

minus008tran

sient

to st

eady

at755

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

m

2

15

1

05

0

minus05

minus1

minus15

minus2Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at815

Hz (

mm

)

0 1 2 3 4 5 6

008

006

004

002

0

minus002

minus004

minus006

minus008Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at875

Hz (

mm

)

0 1 2 3 4 5 6

01

005

0

minus005

minus01Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at775

Hz (

mm

)

0 1 2 3 4 5 6

Time (s)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

0 1 2 3 4 5 6

0 1 2 3 4 5 6

0

008

006

004

002

minus002

minus004

minus006

minus008Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at895

Hz (

mm

)

02

01

0

minus01

minus02

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at835

Hz (

mm

)Figure 12 Theoretical tip displacement responses with third-type mode shape of the beam from transient to steady state

Table 3 The first natural frequencies and the resonant displacements at some certain positions

Resonant frequency or positions The values of frequencies and displacement responses (Hz or mm) ErrorsTheory based on 119880(2)

1(119909) Theory based on 119880(3)

1(119909) Experiment

The 1st natural frequency 815 815 812 037119909 = 364mm 01285 01285 01277 062119909 = 574mm 03022 03022 03001 069119909 = 784mm 05279 05279 05246 063119909 = 994mm 07905 07905 07855 063119909 = 1204mm 10742 10742 10671 066

base excitation using forced vibration theory and the methodof separation of variables A simplified cantilever model byreplacing the tip-mass with an equivalent inertial force andinertial moment acting at the free end of the cantilever beamwas detailed In addition using the Dirac Delta Functionthe base transverse excitation the concentrated equivalentinertia force and inertia moment were represented in theform of distributed loads which can make the method ofseparation of variables be utilized properly In this case sometypical results of natural frequencies and mode shapes of thecantilever beam with a tip-mass were derived using the rein-forced method of separation of variables and the developedforced vibration respectively Furthermore based on modalanalysis method the transient and steady displacement

responses of the cantilever system with a tip-mass and a sinebase excitation were presented in which two different types ofmode shapes calculated by two improved analytical methodswere used Finally an experimental cantilever system wasfabricated and some practical displacement responses weremeasured

From the results of the typical natural frequencies andmode shapes we can easily found that the mode shapesfrom the first improved type were identical to the modeshapes from the second type whether the mode shapes fromthe lower order or from the higher order even the offsetdistance was large The comparing results of the transientand steady displacement responses from the theoreticalanalysis and experimental measurements further proved the

14 Shock and Vibration

005

0

minus0050 2 4 6 8 10 12

0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12

0 2 4 6 8 10 120 2 4 6 8 10 12

0 2 4 6 8 10 12 0 2 4 6 8 10 120 2 4 6 8 10 12

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at732

Hz (

mm

)

0

008

006

004

002

minus002

minus004

minus006

minus008

0

008

006

004

002

minus002

minus004

minus006

minus008

0

008

006

004

002

minus002

minus004

minus006

minus008

0

015

01

005

minus005

minus01

01

005

0

minus005

minus01

2

15

1

05

0

minus05

minus1

minus15

minus2

03

02

01

0

minus01

minus02

03

02

01

0

minus01

minus02

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at752

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at772

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at792

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at812

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at832

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at852

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at872

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at892

Hz (

mm

)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Figure 13 Experimental displacement responses at the position of 119909 = 1204mm from transient to steady state

12

1

08

06

04

02

0120 100 80 60 40 20

78

80

82

84Theo

retic

al st

eady

disp

lace

men

t (m

m)

Position at the beam (mm)Freq

uency

(Hz)

Figure 14 Theoretical steady-displacement responses versus posi-tions and frequencies with mode 119880(2)

1(119909)

improvements of the analytical methods for calculating themode shapes of the cantilever beam with a tip-mass anda base excitation In addition the amplitudes of the modeshapes decreased with the increasing mass value of the tip-mass when the moment of inertia and offset were constantsThe amplitude of the first-order mode shape increased but

12

1

08

06

04

02

0120 100 80 60 40 20

78

80

82

84Theo

retic

al st

eady

disp

lace

men

t (m

m)

Position at the beam (mm)Freq

uency

(Hz)

Figure 15 Theoretical steady-displacement responses versus posi-tions and frequencies with mode 119880(3)

1(119909)

the other nine mode shapes decreased with the increasingmoment of inertia when the mass and offset were set asconstants Conversely the amplitude of the first mode shapedecreased but the other ninemode shapes increased with theincreasing offset distance under the condition that the massand moment of inertia were constants Globally the effect

Shock and Vibration 15

12

1

08

06

04

02

0

Expe

rimen

tal s

tead

y di

spla

cem

ent (

mm

)

120 100 80 60 40 20Position at the beam (mm)

78

80

82

84Freq

uency

(Hz)

Figure 16 Experimental steady-displacement responses versuspositions and frequencies

from varying combinations of the mass and moment inertiaof the tip-mass and the offset distance for the lower-ordermode shapes was stronger than for the higher ones

Furthermore it is clearly noticed that the mass of tip-mass can reduce the values of the natural frequencies andthe increasing mass value gave the decreasing frequencieswhen the moment of inertia of the tip-mass and the offsetdistance were set as constants Similarly the moment ofinertia can further decrease the natural frequencies and thehigher moment of inertia resulted in the lower frequencieswhen the mass and the offset were constants Differently theoffset distance can reduce the first-order natural frequenciesbut increase the other nine order frequencies and the higheroffset distance led to the lower first-order natural frequenciesbut produced the higher other nine order frequencies whenthe mass and the moment of inertia were set to constant Inglobal view the influence from the varying mass momentof inertia and offset distance for the lower-order naturalfrequencies was larger than for the higher-order ones weatherthis influence was negative or positive

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the Natural Science Foundationsof China (no 51275380) and the Scientific and TechnologicalProject of Shaanxi Province (no 2012K06-36) The authorswould like to thank the reviewers for their valuable commentson this paper

References

[1] P A A Laura J L Pombo and E A Susemihl ldquoA note on thevibrations of a clamped-free beam with a mass at the free endrdquoJournal of Sound and Vibration vol 37 no 2 pp 161ndash168 1974

[2] B Rama Bhat andHWagner ldquoNatural frequencies of a uniformcantilever with a tip mass slender in the axial directionrdquo Journalof Sound and Vibration vol 45 no 2 pp 304ndash307 1976

[3] C W S To ldquoVibration of a cantilever beam with a baseexcitation and tip massrdquo Journal of Sound and Vibration vol83 no 4 pp 445ndash460 1982

[4] P A A Laura ldquoComments on lsquoVibration of a cantilever beamwith a base excitation and tip massrsquordquo Journal of Sound andVibration vol 88 no 4 p 569 1983

[5] C W S To ldquoDiscussion on lsquovibration of a cantilever beam witha base excitation and tip massrsquordquo Journal of Sound and Vibrationvol 97 no 2 pp 349ndash351 1984

[6] R G Jacquot ldquoFurther comments on lsquoVibration of a cantileverbeam with a base excitation and tip massrsquordquo Journal of Sound andVibration vol 93 no 2 pp 312ndash313 1984

[7] S AMousavi Lajimi andG R Heppler ldquoComments on lsquonaturalfrequencies of a uniform cantilever with a tip mass slender inthe axial directionrsquordquo Journal of Sound and Vibration vol 331 no12 pp 2964ndash2968 2012

[8] R B Bhat ldquoAuthors response to the comments on lsquonaturalfrequencies of a uniform cantilever with a tip mass slender inthe axial directionrsquordquo Journal of Sound and Vibration vol 331 no12 pp 2962ndash2963 2012

[9] E Esmailzadeh and G Nakhaie-Jazar ldquoPeriodic behavior ofa cantilever beam with end mass subjected to harmonic baseexcitationrdquo International Journal of Non-Linear Mechanics vol33 no 4 pp 567ndash577 1998

[10] H Abramovich and O Hamburger ldquoVibration of a cantilevertimoshenko beam with a tip massrdquo Journal of Sound andVibration vol 148 no 1 pp 162ndash170 1991

[11] B Pratiher ldquoVibration control of a transversely excited can-tilever beam with tip massrdquo Archive of Applied Mechanics vol82 no 1 pp 31ndash42 2012

[12] M Kim M Hoegen J Dugundji and B L Wardle ldquoModelingand experimental verification of proof mass effects on vibrationenergy harvester performancerdquo Smart Materials and Structuresvol 19 Article ID 045023 2010

[13] Y T Li and R Wong ldquoIntegral and series representations ofthe Dirac delta functionrdquo Communications on Pure and AppliedAnalysis vol 7 no 2 pp 229ndash247 2008

[14] M Al-Gwaiz Sturm-LiouvilleTheory and its Applications chap-ter 3 Springer London UK 2008

[15] F Bao-lian ldquoGeneralized reciprocal theorems and their appli-cationsrdquo Applied Mathematics and Mechanics vol 23 no 2 pp203ndash210 2001

[16] C Cai H Zheng M Khan and K HungModeling of MaterialDamping Properties in ANSYS Defense Systems Division Insti-tute of High Performance Computing 2002

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Submit your manuscripts athttpwwwhindawicom

VLSI Design

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Shock and Vibration

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International Journal of

10 Shock and Vibration

where 119876119894(119904) and 119861

119894(119904) are the Laplace transforms of 119902

119894(119905) and

119887(119905) respectively The inverse transformation of the transferfunction will be further obtained as

119892119894(119905)

=

(minus1205962)[119872119880

(119896)

119894119873(119871)minus119903

119888119872(119889119880

(119896)

119894119873(119871) 119889119909)minus119898int

119871

0119880(119896)

119894119873(119909) 119889119909]

120596119894radic1 minus 120577

2

times 119890minus120596119894120577119905 sin(120596

119894radic1 minus 120577

2119905)

(45)

In this case the solutions of (43) can be solved by convolutiontheorem as

119902119894(119905) = 119887 (119905) lowast 119892

119894(119905) = int

119905

0

119887 (120591) 119892119894(119905 minus 120591) 119889120591 (46)

Finally substituting the result of (46) into (38) and after somelengthy algebraic manipulation the absolute displacementresponse of the cantilever beam with tip-mass under basetransverse excitation can be obtained as

119906119886(119909 120596 119905) =

infin

sum

119894=1

119880(119896)

119894119873(119909)

119861 [119872119880(119896)

119894119873(119871) minus 119903

119888119872(119889119880

(119896)

119894119873(119871) 119889119909) minus 119898int

119871

0119880(119896)

119894119873(119909) 119889119909]

radic1205964+ 2 (2120577

2minus 1) 120596

21205962

119894+ 1205964

119894

times [

1205963

120596119894radic1 minus 120577

2

119890minus120596119894120577119905 sin(120596

119894radic1 minus 120577

2119905 + 1206011) + 1205962 sin (120596119905 + 120601

2)]

(47)

where 1206011and 120601

2are determined by tan(120601

1) = 2120577radic1 minus 120577

2

((120596120596119894)2+ 21205772minus 1) and tan(120601

2) = minus2120577(120596120596

119894)(1 minus (120596120596

119894)2)

It can be seen that the solution consists of two parts abouttimeThe first term represents free vibration which will decaywith time exponentially The second term depending on theexcitation represents the forced vibration of the system

42 Experimental Responses and Comparisons with the Theo-retical Results In order to explain the absolute displacementresponse of the cantilever structure under base excitationexperimentally a typical cantilever beam in which theelongated beam was inserted and fixed into the tip-mass tomake sure the gravity center of tip-mass was collinear withthe beam was fabricated (as shown in Figure 9) The chosengeometric and physical properties of the cantilever beamincluding the tip-mass are given in Table 2

The complete experimental devices are shown inFigure 10 The electromagnetic vibrating table type SW-TFAcontrolled by a programmable controller type SW-F-2TF-2was used to create a specific harmonic input excitationThe cantilever beam with a tip-mass was clamped onthe vibrating top platform Eddy-Current Sensors typeDM6008 and ZA210800 were used to measure the inputdisplacement from base vibrating exciter and the absolutedynamic displacement from the beam respectively Allmeasurements from cantilever beam and vibrating exciterwere connected to Digital Oscilloscope type DS1204Band the Industrial Computer including a PC-based DataAcquisitionModule from ADLINK company type DAQ2213The Digital Oscilloscope and Industrial Computer displayedand processed the measurement results using the DAQPilotsoftware and the results from subsequent analysis using theMATLAB software

Table 2 Characteristic properties of the cantilever beam includingtip-mass

Physical properties Beam Tip-massYoungrsquos modulus 119864 (GPa) 69 mdashDensity 120588

119904or 120588119898(kgm3) 2630 7753

Length 119871 or 119871119898(mm) 1204 156

Thickness 119905119904or 119905119898(mm) 49 295

Width 119908 (mm) 102 102Total mass119872 (g)a mdash 3383Offset 119903

119888(mm)b mdash 801

Rotary inertia 119869 (kgsdotm2)b c mdash 333 times 10minus6aThe total mass of the tip-mass including the elongated part of beam wascalculated by119872 = 119871

119898119905119898119908120588119898+ 119871119904119905119904119908(120588119904minus 120588119898) where 119871

119904= 10mm

bThe offset distance and the rotary inertia were calculated according to thegeometry and material properties of the beam and tip-mass using equations119903119888= (120588

1198981199051198981198712

119898minus 1205881198981199051199041198712

119904+ 1205881199041199051199041198712

119904)(2[120588

119898119905119898119871119898+ 119871119904119905119904(120588119904minus 120588119898)])

119869 = 120588119898119871119898119905119898119908 [(112)(119871

2

119898+ 1199052

119898) + ((12)119871

119898minus 119903119888)2] + 119871

119904119905119904119908(120588119904minus

120588119898)[(112)(119871

2

119904+ 1199052

119904) + (119903119888minus (12)119871

119904)2]

cThe corresponding parameters 120601 119896119871 and 119903119888119871 were calculated as follows

120601 = 119872(120588119904119905119904119908) = 21377 119896119871 = radic119869119872119871 = 00824 and 119903

119888119871 = 00665

The analytical model defined by (47) in which two typesof mode shapes derived by two improved analytical methodsare utilized can be used to analyze the steady and transientdisplacement responses of the cantilever beam with a tip-mass and an input base excitation based on the chosenvarying frequencies time and positions along the lengthof the beam Typically the displacement responses frompositions along 119909-axis at 119909

1= 364mm 119909

2= 574mm

1199093

= 784mm 1199094

= 994mm and 1199095

= 1204mmoff the origin point (as shown in Figure 9) were measured

Shock and Vibration 11

y

b(t)

x1

x2

x3

x4

x5

Ls

L

t s

t m x

Lm

rc

Cg

o998400

Figure 9 Typical cantilever beam with a tip-mass

1 2

3

56

7

8

4

Tip-massCantilever beam

(7) Cantilever with a tip-mass

(1) Industrial computer including data acquisitionmodule type DAQ2213

(2) Digital oscilloscope type DS1204B(3) Programmable controller type SW-F-FTF-2(4) Power supply type DP1308A(5) Driver modules for Eddy-current sensors(6) Electromagnetic vibrating table type SW-TFA

(8) Eddy-current sensors type DM6008 and ZA210800

Figure 10 Complete experimental setup

by Eddy-Current Sensors The amplitude of the input sine-displacement excitation for analysis was measured as 119861 =

001mm and the damping ratio for the transverse vibrationaround the fundamental resonant frequency was found to be120577 = 00025 by Logarithmic Decrement Method

In the following paragraphs we focus on the transientand steady responses of the cantilever system with a sine baseexcitation

Figures 11 and 12 show the theoretical displacementresponses at the free end of the beam from transient tosteady state with two improved mode shapes above at thefirst resonant frequency It can be seen obviously that thedisplacement responses with two improved mode shapeswe proposed are almost identical This richly proves thatthe first-type mode shapes reformulated by the reinforcedmethod of separation of variables are equal to the second-type mode shapes derived by the developed method offorced vibration In addition it can be further obtainedthat when the frequency from the base excitation is veryclose but not exactly equal to the resonant frequency of freevibrations of the cantilever system a kind of vibration calledbeating occurs and the period of the beating increases as120596 approached 120596

1 This kind of beating decays exponentially

along the time due to the mechanical dampingFigure 13 shows the experimental displacement responses

at the position of 119909 = 1204mm from transient to steady

state As the same to the theoretical results experimental dis-placement response can be regarded as the sum of waveformsfrom transient and steady states But the transient responsevanished gradually with the time due to the mechanicaldamping and only the steady response left In addition thephenomenon of beating still existed The amplitude of thisbeating phenomenon reduced gradually and the closer theexciting frequency approached to the first natural frequencyof the cantilever system the faster the amplitude of thebeating decreased

Furthermore the amplitudes of the theoreticalwaveformsin Figures 11 and 12 are different from the experimental resultin Figure 13 This may because af the following (a) The effectof mechanical damping with respect to stiffness was onlyconsidered and the mass damping was ignored in theoreticalanalysis which can also weaken the transient vibration faster(b) The first natural frequency of the cantilever system wascalculated and measured by theoretical and experimentalmethods as 815Hz and 812Hz respectively The little dif-ference made that the max amplitudes from theoretical andexperimental results did not appear at the same frequency(c) In theoretical analysis the tip-mass was equated by theinertia force and inertia moment acting at the free end ofthe cantilever so the total length for theoretical analysiswas just the length of the beam of 1204mm Howeverduring experiments the tip-displacement was measured at

12 Shock and Vibration

005

0

minus0050 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6

0 1 2 3 4 5 6 0 1 2 3 4 5 6

0 1 2 3 4 5 6 0 1 2 3 4 5 6

0 1 2 3 4 5 6

0

0

1 2 3 4 5 6

02

01

0

minus01

minus02

02

01

0

0

minus01

minus02

015

01

005

minus005

minus01

008

006

004

002

0

minus002

minus004

minus006

minus008

008

006

004

002

0

minus002

minus004

minus006

minus008

008

006

004

002

minus002

minus004

minus006

minus008

2

15

1

05

0

minus05

minus1

minus15

minus2

01

005

0

minus005

minus01Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at735

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

795

Hz (

mm

)

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at855

Hz (

mm

)

tran

sient

to st

eady

at755

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mTh

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

815

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

875

Hz (

mm

)

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at775

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

835

Hz (

mm

)

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at895

Hz (

mm

)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Figure 11 Theoretical tip displacement responses with the second-type mode shape of the beam from transient to steady state

the approximate position of 119909 = 1204mm This is justan approximate value because Eddy-Current Sensor cannotgive exact measurement results of the displacement at thejunction between the cantilever and the tip-mass This maycause that the amplitude level from theoretical discussionwashigher than from experimental measurement In additionthe amplitude of the instantaneous base displacement fromthe exciter may be not the fixed value 001mm at the startinginstant Therefore the maximum displacement responsesduring the transient state in the experimental measurementsmay shift away from the real experimental results

The theoretical responses with two improved types ofmode shapes and the experimental displacement responsesversus varying positions of beam and the frequencies atsteady state are shown in Figures 14ndash16 respectively Figures14 and 15 show two types of theoretical displacements atthe positions of 119909 = 364mm 574mm 784mm 994mmand 1204mmwith continuous varying frequencies In exper-iments the displacement responses were measured at thesame positions with theoretical calculations but with somecertain frequencies of 782Hz 792Hz 802Hz 807Hz81Hz 812Hz 817Hz 82Hz 83Hz and 84Hz as plottedin Figure 16 It is noticed that when the frequency of baseexcitation approached to the first natural frequency of thecantilever system the amplitude of forced vibration willrapidly increase and become maximum for the case when

the frequency of the excitation coincides with the naturalfrequency Table 3 lists the values of the first natural frequen-cies and the displacement responses from theoretical andexperimental results at some certain positions long the beamunder the first natural frequencies It can be seen that thetheoretical and experimental results are matched very close

Table 3 The first natural frequencies and the resonantdisplacements at some certain positions

In a word the theoretical results with two different modeshapes derived by two improved analytical methods haveextremely the same behaviors at transient and steady stateswith respect to varying frequencies from base excitationThedisplacement responses indicate close agreement betweenthe experimental and analytical results All of these resultsgive full verification for the improvement of the analyticalmethods in this paper utilized in recalculating the modeshapes of a cantilever beam when an offset distance existsbetween the gravity center of the tip-mass and the point ofattachment

5 Conclusions

This paper presents two improved analytical methods ofcalculation for natural frequencies and mode shapes of auniform cantilever beam carrying a tip-mass under an input

Shock and Vibration 13

005

0

minus005Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at735

Hz (

mm

)

0 1 2 3 4 5 6

Time (s)

0 1 2 3 4 5 6

0 1 2 3 4 5 6

02

01

0

minus01

minus02

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at795

Hz (

mm

)

0

015

01

005

minus005

minus01

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at855

Hz (

mm

)

0 1 2 3 4 5 6

Time (s)

008

006

004

002

0

minus002

minus004

minus006

minus008tran

sient

to st

eady

at755

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

m

2

15

1

05

0

minus05

minus1

minus15

minus2Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at815

Hz (

mm

)

0 1 2 3 4 5 6

008

006

004

002

0

minus002

minus004

minus006

minus008Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at875

Hz (

mm

)

0 1 2 3 4 5 6

01

005

0

minus005

minus01Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at775

Hz (

mm

)

0 1 2 3 4 5 6

Time (s)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

0 1 2 3 4 5 6

0 1 2 3 4 5 6

0

008

006

004

002

minus002

minus004

minus006

minus008Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at895

Hz (

mm

)

02

01

0

minus01

minus02

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at835

Hz (

mm

)Figure 12 Theoretical tip displacement responses with third-type mode shape of the beam from transient to steady state

Table 3 The first natural frequencies and the resonant displacements at some certain positions

Resonant frequency or positions The values of frequencies and displacement responses (Hz or mm) ErrorsTheory based on 119880(2)

1(119909) Theory based on 119880(3)

1(119909) Experiment

The 1st natural frequency 815 815 812 037119909 = 364mm 01285 01285 01277 062119909 = 574mm 03022 03022 03001 069119909 = 784mm 05279 05279 05246 063119909 = 994mm 07905 07905 07855 063119909 = 1204mm 10742 10742 10671 066

base excitation using forced vibration theory and the methodof separation of variables A simplified cantilever model byreplacing the tip-mass with an equivalent inertial force andinertial moment acting at the free end of the cantilever beamwas detailed In addition using the Dirac Delta Functionthe base transverse excitation the concentrated equivalentinertia force and inertia moment were represented in theform of distributed loads which can make the method ofseparation of variables be utilized properly In this case sometypical results of natural frequencies and mode shapes of thecantilever beam with a tip-mass were derived using the rein-forced method of separation of variables and the developedforced vibration respectively Furthermore based on modalanalysis method the transient and steady displacement

responses of the cantilever system with a tip-mass and a sinebase excitation were presented in which two different types ofmode shapes calculated by two improved analytical methodswere used Finally an experimental cantilever system wasfabricated and some practical displacement responses weremeasured

From the results of the typical natural frequencies andmode shapes we can easily found that the mode shapesfrom the first improved type were identical to the modeshapes from the second type whether the mode shapes fromthe lower order or from the higher order even the offsetdistance was large The comparing results of the transientand steady displacement responses from the theoreticalanalysis and experimental measurements further proved the

14 Shock and Vibration

005

0

minus0050 2 4 6 8 10 12

0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12

0 2 4 6 8 10 120 2 4 6 8 10 12

0 2 4 6 8 10 12 0 2 4 6 8 10 120 2 4 6 8 10 12

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at732

Hz (

mm

)

0

008

006

004

002

minus002

minus004

minus006

minus008

0

008

006

004

002

minus002

minus004

minus006

minus008

0

008

006

004

002

minus002

minus004

minus006

minus008

0

015

01

005

minus005

minus01

01

005

0

minus005

minus01

2

15

1

05

0

minus05

minus1

minus15

minus2

03

02

01

0

minus01

minus02

03

02

01

0

minus01

minus02

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at752

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at772

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at792

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at812

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at832

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at852

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at872

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at892

Hz (

mm

)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Figure 13 Experimental displacement responses at the position of 119909 = 1204mm from transient to steady state

12

1

08

06

04

02

0120 100 80 60 40 20

78

80

82

84Theo

retic

al st

eady

disp

lace

men

t (m

m)

Position at the beam (mm)Freq

uency

(Hz)

Figure 14 Theoretical steady-displacement responses versus posi-tions and frequencies with mode 119880(2)

1(119909)

improvements of the analytical methods for calculating themode shapes of the cantilever beam with a tip-mass anda base excitation In addition the amplitudes of the modeshapes decreased with the increasing mass value of the tip-mass when the moment of inertia and offset were constantsThe amplitude of the first-order mode shape increased but

12

1

08

06

04

02

0120 100 80 60 40 20

78

80

82

84Theo

retic

al st

eady

disp

lace

men

t (m

m)

Position at the beam (mm)Freq

uency

(Hz)

Figure 15 Theoretical steady-displacement responses versus posi-tions and frequencies with mode 119880(3)

1(119909)

the other nine mode shapes decreased with the increasingmoment of inertia when the mass and offset were set asconstants Conversely the amplitude of the first mode shapedecreased but the other ninemode shapes increased with theincreasing offset distance under the condition that the massand moment of inertia were constants Globally the effect

Shock and Vibration 15

12

1

08

06

04

02

0

Expe

rimen

tal s

tead

y di

spla

cem

ent (

mm

)

120 100 80 60 40 20Position at the beam (mm)

78

80

82

84Freq

uency

(Hz)

Figure 16 Experimental steady-displacement responses versuspositions and frequencies

from varying combinations of the mass and moment inertiaof the tip-mass and the offset distance for the lower-ordermode shapes was stronger than for the higher ones

Furthermore it is clearly noticed that the mass of tip-mass can reduce the values of the natural frequencies andthe increasing mass value gave the decreasing frequencieswhen the moment of inertia of the tip-mass and the offsetdistance were set as constants Similarly the moment ofinertia can further decrease the natural frequencies and thehigher moment of inertia resulted in the lower frequencieswhen the mass and the offset were constants Differently theoffset distance can reduce the first-order natural frequenciesbut increase the other nine order frequencies and the higheroffset distance led to the lower first-order natural frequenciesbut produced the higher other nine order frequencies whenthe mass and the moment of inertia were set to constant Inglobal view the influence from the varying mass momentof inertia and offset distance for the lower-order naturalfrequencies was larger than for the higher-order ones weatherthis influence was negative or positive

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the Natural Science Foundationsof China (no 51275380) and the Scientific and TechnologicalProject of Shaanxi Province (no 2012K06-36) The authorswould like to thank the reviewers for their valuable commentson this paper

References

[1] P A A Laura J L Pombo and E A Susemihl ldquoA note on thevibrations of a clamped-free beam with a mass at the free endrdquoJournal of Sound and Vibration vol 37 no 2 pp 161ndash168 1974

[2] B Rama Bhat andHWagner ldquoNatural frequencies of a uniformcantilever with a tip mass slender in the axial directionrdquo Journalof Sound and Vibration vol 45 no 2 pp 304ndash307 1976

[3] C W S To ldquoVibration of a cantilever beam with a baseexcitation and tip massrdquo Journal of Sound and Vibration vol83 no 4 pp 445ndash460 1982

[4] P A A Laura ldquoComments on lsquoVibration of a cantilever beamwith a base excitation and tip massrsquordquo Journal of Sound andVibration vol 88 no 4 p 569 1983

[5] C W S To ldquoDiscussion on lsquovibration of a cantilever beam witha base excitation and tip massrsquordquo Journal of Sound and Vibrationvol 97 no 2 pp 349ndash351 1984

[6] R G Jacquot ldquoFurther comments on lsquoVibration of a cantileverbeam with a base excitation and tip massrsquordquo Journal of Sound andVibration vol 93 no 2 pp 312ndash313 1984

[7] S AMousavi Lajimi andG R Heppler ldquoComments on lsquonaturalfrequencies of a uniform cantilever with a tip mass slender inthe axial directionrsquordquo Journal of Sound and Vibration vol 331 no12 pp 2964ndash2968 2012

[8] R B Bhat ldquoAuthors response to the comments on lsquonaturalfrequencies of a uniform cantilever with a tip mass slender inthe axial directionrsquordquo Journal of Sound and Vibration vol 331 no12 pp 2962ndash2963 2012

[9] E Esmailzadeh and G Nakhaie-Jazar ldquoPeriodic behavior ofa cantilever beam with end mass subjected to harmonic baseexcitationrdquo International Journal of Non-Linear Mechanics vol33 no 4 pp 567ndash577 1998

[10] H Abramovich and O Hamburger ldquoVibration of a cantilevertimoshenko beam with a tip massrdquo Journal of Sound andVibration vol 148 no 1 pp 162ndash170 1991

[11] B Pratiher ldquoVibration control of a transversely excited can-tilever beam with tip massrdquo Archive of Applied Mechanics vol82 no 1 pp 31ndash42 2012

[12] M Kim M Hoegen J Dugundji and B L Wardle ldquoModelingand experimental verification of proof mass effects on vibrationenergy harvester performancerdquo Smart Materials and Structuresvol 19 Article ID 045023 2010

[13] Y T Li and R Wong ldquoIntegral and series representations ofthe Dirac delta functionrdquo Communications on Pure and AppliedAnalysis vol 7 no 2 pp 229ndash247 2008

[14] M Al-Gwaiz Sturm-LiouvilleTheory and its Applications chap-ter 3 Springer London UK 2008

[15] F Bao-lian ldquoGeneralized reciprocal theorems and their appli-cationsrdquo Applied Mathematics and Mechanics vol 23 no 2 pp203ndash210 2001

[16] C Cai H Zheng M Khan and K HungModeling of MaterialDamping Properties in ANSYS Defense Systems Division Insti-tute of High Performance Computing 2002

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Submit your manuscripts athttpwwwhindawicom

VLSI Design

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Shock and Vibration

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Civil EngineeringAdvances in

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Advances inOptoElectronics

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

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DistributedSensor Networks

International Journal of

Shock and Vibration 11

y

b(t)

x1

x2

x3

x4

x5

Ls

L

t s

t m x

Lm

rc

Cg

o998400

Figure 9 Typical cantilever beam with a tip-mass

1 2

3

56

7

8

4

Tip-massCantilever beam

(7) Cantilever with a tip-mass

(1) Industrial computer including data acquisitionmodule type DAQ2213

(2) Digital oscilloscope type DS1204B(3) Programmable controller type SW-F-FTF-2(4) Power supply type DP1308A(5) Driver modules for Eddy-current sensors(6) Electromagnetic vibrating table type SW-TFA

(8) Eddy-current sensors type DM6008 and ZA210800

Figure 10 Complete experimental setup

by Eddy-Current Sensors The amplitude of the input sine-displacement excitation for analysis was measured as 119861 =

001mm and the damping ratio for the transverse vibrationaround the fundamental resonant frequency was found to be120577 = 00025 by Logarithmic Decrement Method

In the following paragraphs we focus on the transientand steady responses of the cantilever system with a sine baseexcitation

Figures 11 and 12 show the theoretical displacementresponses at the free end of the beam from transient tosteady state with two improved mode shapes above at thefirst resonant frequency It can be seen obviously that thedisplacement responses with two improved mode shapeswe proposed are almost identical This richly proves thatthe first-type mode shapes reformulated by the reinforcedmethod of separation of variables are equal to the second-type mode shapes derived by the developed method offorced vibration In addition it can be further obtainedthat when the frequency from the base excitation is veryclose but not exactly equal to the resonant frequency of freevibrations of the cantilever system a kind of vibration calledbeating occurs and the period of the beating increases as120596 approached 120596

1 This kind of beating decays exponentially

along the time due to the mechanical dampingFigure 13 shows the experimental displacement responses

at the position of 119909 = 1204mm from transient to steady

state As the same to the theoretical results experimental dis-placement response can be regarded as the sum of waveformsfrom transient and steady states But the transient responsevanished gradually with the time due to the mechanicaldamping and only the steady response left In addition thephenomenon of beating still existed The amplitude of thisbeating phenomenon reduced gradually and the closer theexciting frequency approached to the first natural frequencyof the cantilever system the faster the amplitude of thebeating decreased

Furthermore the amplitudes of the theoreticalwaveformsin Figures 11 and 12 are different from the experimental resultin Figure 13 This may because af the following (a) The effectof mechanical damping with respect to stiffness was onlyconsidered and the mass damping was ignored in theoreticalanalysis which can also weaken the transient vibration faster(b) The first natural frequency of the cantilever system wascalculated and measured by theoretical and experimentalmethods as 815Hz and 812Hz respectively The little dif-ference made that the max amplitudes from theoretical andexperimental results did not appear at the same frequency(c) In theoretical analysis the tip-mass was equated by theinertia force and inertia moment acting at the free end ofthe cantilever so the total length for theoretical analysiswas just the length of the beam of 1204mm Howeverduring experiments the tip-displacement was measured at

12 Shock and Vibration

005

0

minus0050 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6

0 1 2 3 4 5 6 0 1 2 3 4 5 6

0 1 2 3 4 5 6 0 1 2 3 4 5 6

0 1 2 3 4 5 6

0

0

1 2 3 4 5 6

02

01

0

minus01

minus02

02

01

0

0

minus01

minus02

015

01

005

minus005

minus01

008

006

004

002

0

minus002

minus004

minus006

minus008

008

006

004

002

0

minus002

minus004

minus006

minus008

008

006

004

002

minus002

minus004

minus006

minus008

2

15

1

05

0

minus05

minus1

minus15

minus2

01

005

0

minus005

minus01Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at735

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

795

Hz (

mm

)

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at855

Hz (

mm

)

tran

sient

to st

eady

at755

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mTh

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

815

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

875

Hz (

mm

)

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at775

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

835

Hz (

mm

)

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at895

Hz (

mm

)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Figure 11 Theoretical tip displacement responses with the second-type mode shape of the beam from transient to steady state

the approximate position of 119909 = 1204mm This is justan approximate value because Eddy-Current Sensor cannotgive exact measurement results of the displacement at thejunction between the cantilever and the tip-mass This maycause that the amplitude level from theoretical discussionwashigher than from experimental measurement In additionthe amplitude of the instantaneous base displacement fromthe exciter may be not the fixed value 001mm at the startinginstant Therefore the maximum displacement responsesduring the transient state in the experimental measurementsmay shift away from the real experimental results

The theoretical responses with two improved types ofmode shapes and the experimental displacement responsesversus varying positions of beam and the frequencies atsteady state are shown in Figures 14ndash16 respectively Figures14 and 15 show two types of theoretical displacements atthe positions of 119909 = 364mm 574mm 784mm 994mmand 1204mmwith continuous varying frequencies In exper-iments the displacement responses were measured at thesame positions with theoretical calculations but with somecertain frequencies of 782Hz 792Hz 802Hz 807Hz81Hz 812Hz 817Hz 82Hz 83Hz and 84Hz as plottedin Figure 16 It is noticed that when the frequency of baseexcitation approached to the first natural frequency of thecantilever system the amplitude of forced vibration willrapidly increase and become maximum for the case when

the frequency of the excitation coincides with the naturalfrequency Table 3 lists the values of the first natural frequen-cies and the displacement responses from theoretical andexperimental results at some certain positions long the beamunder the first natural frequencies It can be seen that thetheoretical and experimental results are matched very close

Table 3 The first natural frequencies and the resonantdisplacements at some certain positions

In a word the theoretical results with two different modeshapes derived by two improved analytical methods haveextremely the same behaviors at transient and steady stateswith respect to varying frequencies from base excitationThedisplacement responses indicate close agreement betweenthe experimental and analytical results All of these resultsgive full verification for the improvement of the analyticalmethods in this paper utilized in recalculating the modeshapes of a cantilever beam when an offset distance existsbetween the gravity center of the tip-mass and the point ofattachment

5 Conclusions

This paper presents two improved analytical methods ofcalculation for natural frequencies and mode shapes of auniform cantilever beam carrying a tip-mass under an input

Shock and Vibration 13

005

0

minus005Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at735

Hz (

mm

)

0 1 2 3 4 5 6

Time (s)

0 1 2 3 4 5 6

0 1 2 3 4 5 6

02

01

0

minus01

minus02

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at795

Hz (

mm

)

0

015

01

005

minus005

minus01

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at855

Hz (

mm

)

0 1 2 3 4 5 6

Time (s)

008

006

004

002

0

minus002

minus004

minus006

minus008tran

sient

to st

eady

at755

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

m

2

15

1

05

0

minus05

minus1

minus15

minus2Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at815

Hz (

mm

)

0 1 2 3 4 5 6

008

006

004

002

0

minus002

minus004

minus006

minus008Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at875

Hz (

mm

)

0 1 2 3 4 5 6

01

005

0

minus005

minus01Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at775

Hz (

mm

)

0 1 2 3 4 5 6

Time (s)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

0 1 2 3 4 5 6

0 1 2 3 4 5 6

0

008

006

004

002

minus002

minus004

minus006

minus008Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at895

Hz (

mm

)

02

01

0

minus01

minus02

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at835

Hz (

mm

)Figure 12 Theoretical tip displacement responses with third-type mode shape of the beam from transient to steady state

Table 3 The first natural frequencies and the resonant displacements at some certain positions

Resonant frequency or positions The values of frequencies and displacement responses (Hz or mm) ErrorsTheory based on 119880(2)

1(119909) Theory based on 119880(3)

1(119909) Experiment

The 1st natural frequency 815 815 812 037119909 = 364mm 01285 01285 01277 062119909 = 574mm 03022 03022 03001 069119909 = 784mm 05279 05279 05246 063119909 = 994mm 07905 07905 07855 063119909 = 1204mm 10742 10742 10671 066

base excitation using forced vibration theory and the methodof separation of variables A simplified cantilever model byreplacing the tip-mass with an equivalent inertial force andinertial moment acting at the free end of the cantilever beamwas detailed In addition using the Dirac Delta Functionthe base transverse excitation the concentrated equivalentinertia force and inertia moment were represented in theform of distributed loads which can make the method ofseparation of variables be utilized properly In this case sometypical results of natural frequencies and mode shapes of thecantilever beam with a tip-mass were derived using the rein-forced method of separation of variables and the developedforced vibration respectively Furthermore based on modalanalysis method the transient and steady displacement

responses of the cantilever system with a tip-mass and a sinebase excitation were presented in which two different types ofmode shapes calculated by two improved analytical methodswere used Finally an experimental cantilever system wasfabricated and some practical displacement responses weremeasured

From the results of the typical natural frequencies andmode shapes we can easily found that the mode shapesfrom the first improved type were identical to the modeshapes from the second type whether the mode shapes fromthe lower order or from the higher order even the offsetdistance was large The comparing results of the transientand steady displacement responses from the theoreticalanalysis and experimental measurements further proved the

14 Shock and Vibration

005

0

minus0050 2 4 6 8 10 12

0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12

0 2 4 6 8 10 120 2 4 6 8 10 12

0 2 4 6 8 10 12 0 2 4 6 8 10 120 2 4 6 8 10 12

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at732

Hz (

mm

)

0

008

006

004

002

minus002

minus004

minus006

minus008

0

008

006

004

002

minus002

minus004

minus006

minus008

0

008

006

004

002

minus002

minus004

minus006

minus008

0

015

01

005

minus005

minus01

01

005

0

minus005

minus01

2

15

1

05

0

minus05

minus1

minus15

minus2

03

02

01

0

minus01

minus02

03

02

01

0

minus01

minus02

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at752

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at772

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at792

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at812

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at832

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at852

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at872

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at892

Hz (

mm

)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Figure 13 Experimental displacement responses at the position of 119909 = 1204mm from transient to steady state

12

1

08

06

04

02

0120 100 80 60 40 20

78

80

82

84Theo

retic

al st

eady

disp

lace

men

t (m

m)

Position at the beam (mm)Freq

uency

(Hz)

Figure 14 Theoretical steady-displacement responses versus posi-tions and frequencies with mode 119880(2)

1(119909)

improvements of the analytical methods for calculating themode shapes of the cantilever beam with a tip-mass anda base excitation In addition the amplitudes of the modeshapes decreased with the increasing mass value of the tip-mass when the moment of inertia and offset were constantsThe amplitude of the first-order mode shape increased but

12

1

08

06

04

02

0120 100 80 60 40 20

78

80

82

84Theo

retic

al st

eady

disp

lace

men

t (m

m)

Position at the beam (mm)Freq

uency

(Hz)

Figure 15 Theoretical steady-displacement responses versus posi-tions and frequencies with mode 119880(3)

1(119909)

the other nine mode shapes decreased with the increasingmoment of inertia when the mass and offset were set asconstants Conversely the amplitude of the first mode shapedecreased but the other ninemode shapes increased with theincreasing offset distance under the condition that the massand moment of inertia were constants Globally the effect

Shock and Vibration 15

12

1

08

06

04

02

0

Expe

rimen

tal s

tead

y di

spla

cem

ent (

mm

)

120 100 80 60 40 20Position at the beam (mm)

78

80

82

84Freq

uency

(Hz)

Figure 16 Experimental steady-displacement responses versuspositions and frequencies

from varying combinations of the mass and moment inertiaof the tip-mass and the offset distance for the lower-ordermode shapes was stronger than for the higher ones

Furthermore it is clearly noticed that the mass of tip-mass can reduce the values of the natural frequencies andthe increasing mass value gave the decreasing frequencieswhen the moment of inertia of the tip-mass and the offsetdistance were set as constants Similarly the moment ofinertia can further decrease the natural frequencies and thehigher moment of inertia resulted in the lower frequencieswhen the mass and the offset were constants Differently theoffset distance can reduce the first-order natural frequenciesbut increase the other nine order frequencies and the higheroffset distance led to the lower first-order natural frequenciesbut produced the higher other nine order frequencies whenthe mass and the moment of inertia were set to constant Inglobal view the influence from the varying mass momentof inertia and offset distance for the lower-order naturalfrequencies was larger than for the higher-order ones weatherthis influence was negative or positive

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the Natural Science Foundationsof China (no 51275380) and the Scientific and TechnologicalProject of Shaanxi Province (no 2012K06-36) The authorswould like to thank the reviewers for their valuable commentson this paper

References

[1] P A A Laura J L Pombo and E A Susemihl ldquoA note on thevibrations of a clamped-free beam with a mass at the free endrdquoJournal of Sound and Vibration vol 37 no 2 pp 161ndash168 1974

[2] B Rama Bhat andHWagner ldquoNatural frequencies of a uniformcantilever with a tip mass slender in the axial directionrdquo Journalof Sound and Vibration vol 45 no 2 pp 304ndash307 1976

[3] C W S To ldquoVibration of a cantilever beam with a baseexcitation and tip massrdquo Journal of Sound and Vibration vol83 no 4 pp 445ndash460 1982

[4] P A A Laura ldquoComments on lsquoVibration of a cantilever beamwith a base excitation and tip massrsquordquo Journal of Sound andVibration vol 88 no 4 p 569 1983

[5] C W S To ldquoDiscussion on lsquovibration of a cantilever beam witha base excitation and tip massrsquordquo Journal of Sound and Vibrationvol 97 no 2 pp 349ndash351 1984

[6] R G Jacquot ldquoFurther comments on lsquoVibration of a cantileverbeam with a base excitation and tip massrsquordquo Journal of Sound andVibration vol 93 no 2 pp 312ndash313 1984

[7] S AMousavi Lajimi andG R Heppler ldquoComments on lsquonaturalfrequencies of a uniform cantilever with a tip mass slender inthe axial directionrsquordquo Journal of Sound and Vibration vol 331 no12 pp 2964ndash2968 2012

[8] R B Bhat ldquoAuthors response to the comments on lsquonaturalfrequencies of a uniform cantilever with a tip mass slender inthe axial directionrsquordquo Journal of Sound and Vibration vol 331 no12 pp 2962ndash2963 2012

[9] E Esmailzadeh and G Nakhaie-Jazar ldquoPeriodic behavior ofa cantilever beam with end mass subjected to harmonic baseexcitationrdquo International Journal of Non-Linear Mechanics vol33 no 4 pp 567ndash577 1998

[10] H Abramovich and O Hamburger ldquoVibration of a cantilevertimoshenko beam with a tip massrdquo Journal of Sound andVibration vol 148 no 1 pp 162ndash170 1991

[11] B Pratiher ldquoVibration control of a transversely excited can-tilever beam with tip massrdquo Archive of Applied Mechanics vol82 no 1 pp 31ndash42 2012

[12] M Kim M Hoegen J Dugundji and B L Wardle ldquoModelingand experimental verification of proof mass effects on vibrationenergy harvester performancerdquo Smart Materials and Structuresvol 19 Article ID 045023 2010

[13] Y T Li and R Wong ldquoIntegral and series representations ofthe Dirac delta functionrdquo Communications on Pure and AppliedAnalysis vol 7 no 2 pp 229ndash247 2008

[14] M Al-Gwaiz Sturm-LiouvilleTheory and its Applications chap-ter 3 Springer London UK 2008

[15] F Bao-lian ldquoGeneralized reciprocal theorems and their appli-cationsrdquo Applied Mathematics and Mechanics vol 23 no 2 pp203ndash210 2001

[16] C Cai H Zheng M Khan and K HungModeling of MaterialDamping Properties in ANSYS Defense Systems Division Insti-tute of High Performance Computing 2002

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

12 Shock and Vibration

005

0

minus0050 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6

0 1 2 3 4 5 6 0 1 2 3 4 5 6

0 1 2 3 4 5 6 0 1 2 3 4 5 6

0 1 2 3 4 5 6

0

0

1 2 3 4 5 6

02

01

0

minus01

minus02

02

01

0

0

minus01

minus02

015

01

005

minus005

minus01

008

006

004

002

0

minus002

minus004

minus006

minus008

008

006

004

002

0

minus002

minus004

minus006

minus008

008

006

004

002

minus002

minus004

minus006

minus008

2

15

1

05

0

minus05

minus1

minus15

minus2

01

005

0

minus005

minus01Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at735

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

795

Hz (

mm

)

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at855

Hz (

mm

)

tran

sient

to st

eady

at755

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mTh

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

815

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

875

Hz (

mm

)

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at775

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

mtr

ansie

nt to

stea

dy at

835

Hz (

mm

)

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at895

Hz (

mm

)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Figure 11 Theoretical tip displacement responses with the second-type mode shape of the beam from transient to steady state

the approximate position of 119909 = 1204mm This is justan approximate value because Eddy-Current Sensor cannotgive exact measurement results of the displacement at thejunction between the cantilever and the tip-mass This maycause that the amplitude level from theoretical discussionwashigher than from experimental measurement In additionthe amplitude of the instantaneous base displacement fromthe exciter may be not the fixed value 001mm at the startinginstant Therefore the maximum displacement responsesduring the transient state in the experimental measurementsmay shift away from the real experimental results

The theoretical responses with two improved types ofmode shapes and the experimental displacement responsesversus varying positions of beam and the frequencies atsteady state are shown in Figures 14ndash16 respectively Figures14 and 15 show two types of theoretical displacements atthe positions of 119909 = 364mm 574mm 784mm 994mmand 1204mmwith continuous varying frequencies In exper-iments the displacement responses were measured at thesame positions with theoretical calculations but with somecertain frequencies of 782Hz 792Hz 802Hz 807Hz81Hz 812Hz 817Hz 82Hz 83Hz and 84Hz as plottedin Figure 16 It is noticed that when the frequency of baseexcitation approached to the first natural frequency of thecantilever system the amplitude of forced vibration willrapidly increase and become maximum for the case when

the frequency of the excitation coincides with the naturalfrequency Table 3 lists the values of the first natural frequen-cies and the displacement responses from theoretical andexperimental results at some certain positions long the beamunder the first natural frequencies It can be seen that thetheoretical and experimental results are matched very close

Table 3 The first natural frequencies and the resonantdisplacements at some certain positions

In a word the theoretical results with two different modeshapes derived by two improved analytical methods haveextremely the same behaviors at transient and steady stateswith respect to varying frequencies from base excitationThedisplacement responses indicate close agreement betweenthe experimental and analytical results All of these resultsgive full verification for the improvement of the analyticalmethods in this paper utilized in recalculating the modeshapes of a cantilever beam when an offset distance existsbetween the gravity center of the tip-mass and the point ofattachment

5 Conclusions

This paper presents two improved analytical methods ofcalculation for natural frequencies and mode shapes of auniform cantilever beam carrying a tip-mass under an input

Shock and Vibration 13

005

0

minus005Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at735

Hz (

mm

)

0 1 2 3 4 5 6

Time (s)

0 1 2 3 4 5 6

0 1 2 3 4 5 6

02

01

0

minus01

minus02

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at795

Hz (

mm

)

0

015

01

005

minus005

minus01

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at855

Hz (

mm

)

0 1 2 3 4 5 6

Time (s)

008

006

004

002

0

minus002

minus004

minus006

minus008tran

sient

to st

eady

at755

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

m

2

15

1

05

0

minus05

minus1

minus15

minus2Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at815

Hz (

mm

)

0 1 2 3 4 5 6

008

006

004

002

0

minus002

minus004

minus006

minus008Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at875

Hz (

mm

)

0 1 2 3 4 5 6

01

005

0

minus005

minus01Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at775

Hz (

mm

)

0 1 2 3 4 5 6

Time (s)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

0 1 2 3 4 5 6

0 1 2 3 4 5 6

0

008

006

004

002

minus002

minus004

minus006

minus008Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at895

Hz (

mm

)

02

01

0

minus01

minus02

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at835

Hz (

mm

)Figure 12 Theoretical tip displacement responses with third-type mode shape of the beam from transient to steady state

Table 3 The first natural frequencies and the resonant displacements at some certain positions

Resonant frequency or positions The values of frequencies and displacement responses (Hz or mm) ErrorsTheory based on 119880(2)

1(119909) Theory based on 119880(3)

1(119909) Experiment

The 1st natural frequency 815 815 812 037119909 = 364mm 01285 01285 01277 062119909 = 574mm 03022 03022 03001 069119909 = 784mm 05279 05279 05246 063119909 = 994mm 07905 07905 07855 063119909 = 1204mm 10742 10742 10671 066

base excitation using forced vibration theory and the methodof separation of variables A simplified cantilever model byreplacing the tip-mass with an equivalent inertial force andinertial moment acting at the free end of the cantilever beamwas detailed In addition using the Dirac Delta Functionthe base transverse excitation the concentrated equivalentinertia force and inertia moment were represented in theform of distributed loads which can make the method ofseparation of variables be utilized properly In this case sometypical results of natural frequencies and mode shapes of thecantilever beam with a tip-mass were derived using the rein-forced method of separation of variables and the developedforced vibration respectively Furthermore based on modalanalysis method the transient and steady displacement

responses of the cantilever system with a tip-mass and a sinebase excitation were presented in which two different types ofmode shapes calculated by two improved analytical methodswere used Finally an experimental cantilever system wasfabricated and some practical displacement responses weremeasured

From the results of the typical natural frequencies andmode shapes we can easily found that the mode shapesfrom the first improved type were identical to the modeshapes from the second type whether the mode shapes fromthe lower order or from the higher order even the offsetdistance was large The comparing results of the transientand steady displacement responses from the theoreticalanalysis and experimental measurements further proved the

14 Shock and Vibration

005

0

minus0050 2 4 6 8 10 12

0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12

0 2 4 6 8 10 120 2 4 6 8 10 12

0 2 4 6 8 10 12 0 2 4 6 8 10 120 2 4 6 8 10 12

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at732

Hz (

mm

)

0

008

006

004

002

minus002

minus004

minus006

minus008

0

008

006

004

002

minus002

minus004

minus006

minus008

0

008

006

004

002

minus002

minus004

minus006

minus008

0

015

01

005

minus005

minus01

01

005

0

minus005

minus01

2

15

1

05

0

minus05

minus1

minus15

minus2

03

02

01

0

minus01

minus02

03

02

01

0

minus01

minus02

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at752

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at772

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at792

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at812

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at832

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at852

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at872

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at892

Hz (

mm

)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Figure 13 Experimental displacement responses at the position of 119909 = 1204mm from transient to steady state

12

1

08

06

04

02

0120 100 80 60 40 20

78

80

82

84Theo

retic

al st

eady

disp

lace

men

t (m

m)

Position at the beam (mm)Freq

uency

(Hz)

Figure 14 Theoretical steady-displacement responses versus posi-tions and frequencies with mode 119880(2)

1(119909)

improvements of the analytical methods for calculating themode shapes of the cantilever beam with a tip-mass anda base excitation In addition the amplitudes of the modeshapes decreased with the increasing mass value of the tip-mass when the moment of inertia and offset were constantsThe amplitude of the first-order mode shape increased but

12

1

08

06

04

02

0120 100 80 60 40 20

78

80

82

84Theo

retic

al st

eady

disp

lace

men

t (m

m)

Position at the beam (mm)Freq

uency

(Hz)

Figure 15 Theoretical steady-displacement responses versus posi-tions and frequencies with mode 119880(3)

1(119909)

the other nine mode shapes decreased with the increasingmoment of inertia when the mass and offset were set asconstants Conversely the amplitude of the first mode shapedecreased but the other ninemode shapes increased with theincreasing offset distance under the condition that the massand moment of inertia were constants Globally the effect

Shock and Vibration 15

12

1

08

06

04

02

0

Expe

rimen

tal s

tead

y di

spla

cem

ent (

mm

)

120 100 80 60 40 20Position at the beam (mm)

78

80

82

84Freq

uency

(Hz)

Figure 16 Experimental steady-displacement responses versuspositions and frequencies

from varying combinations of the mass and moment inertiaof the tip-mass and the offset distance for the lower-ordermode shapes was stronger than for the higher ones

Furthermore it is clearly noticed that the mass of tip-mass can reduce the values of the natural frequencies andthe increasing mass value gave the decreasing frequencieswhen the moment of inertia of the tip-mass and the offsetdistance were set as constants Similarly the moment ofinertia can further decrease the natural frequencies and thehigher moment of inertia resulted in the lower frequencieswhen the mass and the offset were constants Differently theoffset distance can reduce the first-order natural frequenciesbut increase the other nine order frequencies and the higheroffset distance led to the lower first-order natural frequenciesbut produced the higher other nine order frequencies whenthe mass and the moment of inertia were set to constant Inglobal view the influence from the varying mass momentof inertia and offset distance for the lower-order naturalfrequencies was larger than for the higher-order ones weatherthis influence was negative or positive

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the Natural Science Foundationsof China (no 51275380) and the Scientific and TechnologicalProject of Shaanxi Province (no 2012K06-36) The authorswould like to thank the reviewers for their valuable commentson this paper

References

[1] P A A Laura J L Pombo and E A Susemihl ldquoA note on thevibrations of a clamped-free beam with a mass at the free endrdquoJournal of Sound and Vibration vol 37 no 2 pp 161ndash168 1974

[2] B Rama Bhat andHWagner ldquoNatural frequencies of a uniformcantilever with a tip mass slender in the axial directionrdquo Journalof Sound and Vibration vol 45 no 2 pp 304ndash307 1976

[3] C W S To ldquoVibration of a cantilever beam with a baseexcitation and tip massrdquo Journal of Sound and Vibration vol83 no 4 pp 445ndash460 1982

[4] P A A Laura ldquoComments on lsquoVibration of a cantilever beamwith a base excitation and tip massrsquordquo Journal of Sound andVibration vol 88 no 4 p 569 1983

[5] C W S To ldquoDiscussion on lsquovibration of a cantilever beam witha base excitation and tip massrsquordquo Journal of Sound and Vibrationvol 97 no 2 pp 349ndash351 1984

[6] R G Jacquot ldquoFurther comments on lsquoVibration of a cantileverbeam with a base excitation and tip massrsquordquo Journal of Sound andVibration vol 93 no 2 pp 312ndash313 1984

[7] S AMousavi Lajimi andG R Heppler ldquoComments on lsquonaturalfrequencies of a uniform cantilever with a tip mass slender inthe axial directionrsquordquo Journal of Sound and Vibration vol 331 no12 pp 2964ndash2968 2012

[8] R B Bhat ldquoAuthors response to the comments on lsquonaturalfrequencies of a uniform cantilever with a tip mass slender inthe axial directionrsquordquo Journal of Sound and Vibration vol 331 no12 pp 2962ndash2963 2012

[9] E Esmailzadeh and G Nakhaie-Jazar ldquoPeriodic behavior ofa cantilever beam with end mass subjected to harmonic baseexcitationrdquo International Journal of Non-Linear Mechanics vol33 no 4 pp 567ndash577 1998

[10] H Abramovich and O Hamburger ldquoVibration of a cantilevertimoshenko beam with a tip massrdquo Journal of Sound andVibration vol 148 no 1 pp 162ndash170 1991

[11] B Pratiher ldquoVibration control of a transversely excited can-tilever beam with tip massrdquo Archive of Applied Mechanics vol82 no 1 pp 31ndash42 2012

[12] M Kim M Hoegen J Dugundji and B L Wardle ldquoModelingand experimental verification of proof mass effects on vibrationenergy harvester performancerdquo Smart Materials and Structuresvol 19 Article ID 045023 2010

[13] Y T Li and R Wong ldquoIntegral and series representations ofthe Dirac delta functionrdquo Communications on Pure and AppliedAnalysis vol 7 no 2 pp 229ndash247 2008

[14] M Al-Gwaiz Sturm-LiouvilleTheory and its Applications chap-ter 3 Springer London UK 2008

[15] F Bao-lian ldquoGeneralized reciprocal theorems and their appli-cationsrdquo Applied Mathematics and Mechanics vol 23 no 2 pp203ndash210 2001

[16] C Cai H Zheng M Khan and K HungModeling of MaterialDamping Properties in ANSYS Defense Systems Division Insti-tute of High Performance Computing 2002

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 13

005

0

minus005Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at735

Hz (

mm

)

0 1 2 3 4 5 6

Time (s)

0 1 2 3 4 5 6

0 1 2 3 4 5 6

02

01

0

minus01

minus02

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at795

Hz (

mm

)

0

015

01

005

minus005

minus01

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at855

Hz (

mm

)

0 1 2 3 4 5 6

Time (s)

008

006

004

002

0

minus002

minus004

minus006

minus008tran

sient

to st

eady

at755

Hz (

mm

)Th

eore

tical

tip-

disp

lace

men

t fro

m

2

15

1

05

0

minus05

minus1

minus15

minus2Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at815

Hz (

mm

)

0 1 2 3 4 5 6

008

006

004

002

0

minus002

minus004

minus006

minus008Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at875

Hz (

mm

)

0 1 2 3 4 5 6

01

005

0

minus005

minus01Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at775

Hz (

mm

)

0 1 2 3 4 5 6

Time (s)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

0 1 2 3 4 5 6

0 1 2 3 4 5 6

0

008

006

004

002

minus002

minus004

minus006

minus008Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at895

Hz (

mm

)

02

01

0

minus01

minus02

Theo

retic

al ti

p-di

spla

cem

ent f

rom

tran

sient

to st

eady

at835

Hz (

mm

)Figure 12 Theoretical tip displacement responses with third-type mode shape of the beam from transient to steady state

Table 3 The first natural frequencies and the resonant displacements at some certain positions

Resonant frequency or positions The values of frequencies and displacement responses (Hz or mm) ErrorsTheory based on 119880(2)

1(119909) Theory based on 119880(3)

1(119909) Experiment

The 1st natural frequency 815 815 812 037119909 = 364mm 01285 01285 01277 062119909 = 574mm 03022 03022 03001 069119909 = 784mm 05279 05279 05246 063119909 = 994mm 07905 07905 07855 063119909 = 1204mm 10742 10742 10671 066

base excitation using forced vibration theory and the methodof separation of variables A simplified cantilever model byreplacing the tip-mass with an equivalent inertial force andinertial moment acting at the free end of the cantilever beamwas detailed In addition using the Dirac Delta Functionthe base transverse excitation the concentrated equivalentinertia force and inertia moment were represented in theform of distributed loads which can make the method ofseparation of variables be utilized properly In this case sometypical results of natural frequencies and mode shapes of thecantilever beam with a tip-mass were derived using the rein-forced method of separation of variables and the developedforced vibration respectively Furthermore based on modalanalysis method the transient and steady displacement

responses of the cantilever system with a tip-mass and a sinebase excitation were presented in which two different types ofmode shapes calculated by two improved analytical methodswere used Finally an experimental cantilever system wasfabricated and some practical displacement responses weremeasured

From the results of the typical natural frequencies andmode shapes we can easily found that the mode shapesfrom the first improved type were identical to the modeshapes from the second type whether the mode shapes fromthe lower order or from the higher order even the offsetdistance was large The comparing results of the transientand steady displacement responses from the theoreticalanalysis and experimental measurements further proved the

14 Shock and Vibration

005

0

minus0050 2 4 6 8 10 12

0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12

0 2 4 6 8 10 120 2 4 6 8 10 12

0 2 4 6 8 10 12 0 2 4 6 8 10 120 2 4 6 8 10 12

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at732

Hz (

mm

)

0

008

006

004

002

minus002

minus004

minus006

minus008

0

008

006

004

002

minus002

minus004

minus006

minus008

0

008

006

004

002

minus002

minus004

minus006

minus008

0

015

01

005

minus005

minus01

01

005

0

minus005

minus01

2

15

1

05

0

minus05

minus1

minus15

minus2

03

02

01

0

minus01

minus02

03

02

01

0

minus01

minus02

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at752

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at772

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at792

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at812

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at832

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at852

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at872

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at892

Hz (

mm

)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Figure 13 Experimental displacement responses at the position of 119909 = 1204mm from transient to steady state

12

1

08

06

04

02

0120 100 80 60 40 20

78

80

82

84Theo

retic

al st

eady

disp

lace

men

t (m

m)

Position at the beam (mm)Freq

uency

(Hz)

Figure 14 Theoretical steady-displacement responses versus posi-tions and frequencies with mode 119880(2)

1(119909)

improvements of the analytical methods for calculating themode shapes of the cantilever beam with a tip-mass anda base excitation In addition the amplitudes of the modeshapes decreased with the increasing mass value of the tip-mass when the moment of inertia and offset were constantsThe amplitude of the first-order mode shape increased but

12

1

08

06

04

02

0120 100 80 60 40 20

78

80

82

84Theo

retic

al st

eady

disp

lace

men

t (m

m)

Position at the beam (mm)Freq

uency

(Hz)

Figure 15 Theoretical steady-displacement responses versus posi-tions and frequencies with mode 119880(3)

1(119909)

the other nine mode shapes decreased with the increasingmoment of inertia when the mass and offset were set asconstants Conversely the amplitude of the first mode shapedecreased but the other ninemode shapes increased with theincreasing offset distance under the condition that the massand moment of inertia were constants Globally the effect

Shock and Vibration 15

12

1

08

06

04

02

0

Expe

rimen

tal s

tead

y di

spla

cem

ent (

mm

)

120 100 80 60 40 20Position at the beam (mm)

78

80

82

84Freq

uency

(Hz)

Figure 16 Experimental steady-displacement responses versuspositions and frequencies

from varying combinations of the mass and moment inertiaof the tip-mass and the offset distance for the lower-ordermode shapes was stronger than for the higher ones

Furthermore it is clearly noticed that the mass of tip-mass can reduce the values of the natural frequencies andthe increasing mass value gave the decreasing frequencieswhen the moment of inertia of the tip-mass and the offsetdistance were set as constants Similarly the moment ofinertia can further decrease the natural frequencies and thehigher moment of inertia resulted in the lower frequencieswhen the mass and the offset were constants Differently theoffset distance can reduce the first-order natural frequenciesbut increase the other nine order frequencies and the higheroffset distance led to the lower first-order natural frequenciesbut produced the higher other nine order frequencies whenthe mass and the moment of inertia were set to constant Inglobal view the influence from the varying mass momentof inertia and offset distance for the lower-order naturalfrequencies was larger than for the higher-order ones weatherthis influence was negative or positive

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the Natural Science Foundationsof China (no 51275380) and the Scientific and TechnologicalProject of Shaanxi Province (no 2012K06-36) The authorswould like to thank the reviewers for their valuable commentson this paper

References

[1] P A A Laura J L Pombo and E A Susemihl ldquoA note on thevibrations of a clamped-free beam with a mass at the free endrdquoJournal of Sound and Vibration vol 37 no 2 pp 161ndash168 1974

[2] B Rama Bhat andHWagner ldquoNatural frequencies of a uniformcantilever with a tip mass slender in the axial directionrdquo Journalof Sound and Vibration vol 45 no 2 pp 304ndash307 1976

[3] C W S To ldquoVibration of a cantilever beam with a baseexcitation and tip massrdquo Journal of Sound and Vibration vol83 no 4 pp 445ndash460 1982

[4] P A A Laura ldquoComments on lsquoVibration of a cantilever beamwith a base excitation and tip massrsquordquo Journal of Sound andVibration vol 88 no 4 p 569 1983

[5] C W S To ldquoDiscussion on lsquovibration of a cantilever beam witha base excitation and tip massrsquordquo Journal of Sound and Vibrationvol 97 no 2 pp 349ndash351 1984

[6] R G Jacquot ldquoFurther comments on lsquoVibration of a cantileverbeam with a base excitation and tip massrsquordquo Journal of Sound andVibration vol 93 no 2 pp 312ndash313 1984

[7] S AMousavi Lajimi andG R Heppler ldquoComments on lsquonaturalfrequencies of a uniform cantilever with a tip mass slender inthe axial directionrsquordquo Journal of Sound and Vibration vol 331 no12 pp 2964ndash2968 2012

[8] R B Bhat ldquoAuthors response to the comments on lsquonaturalfrequencies of a uniform cantilever with a tip mass slender inthe axial directionrsquordquo Journal of Sound and Vibration vol 331 no12 pp 2962ndash2963 2012

[9] E Esmailzadeh and G Nakhaie-Jazar ldquoPeriodic behavior ofa cantilever beam with end mass subjected to harmonic baseexcitationrdquo International Journal of Non-Linear Mechanics vol33 no 4 pp 567ndash577 1998

[10] H Abramovich and O Hamburger ldquoVibration of a cantilevertimoshenko beam with a tip massrdquo Journal of Sound andVibration vol 148 no 1 pp 162ndash170 1991

[11] B Pratiher ldquoVibration control of a transversely excited can-tilever beam with tip massrdquo Archive of Applied Mechanics vol82 no 1 pp 31ndash42 2012

[12] M Kim M Hoegen J Dugundji and B L Wardle ldquoModelingand experimental verification of proof mass effects on vibrationenergy harvester performancerdquo Smart Materials and Structuresvol 19 Article ID 045023 2010

[13] Y T Li and R Wong ldquoIntegral and series representations ofthe Dirac delta functionrdquo Communications on Pure and AppliedAnalysis vol 7 no 2 pp 229ndash247 2008

[14] M Al-Gwaiz Sturm-LiouvilleTheory and its Applications chap-ter 3 Springer London UK 2008

[15] F Bao-lian ldquoGeneralized reciprocal theorems and their appli-cationsrdquo Applied Mathematics and Mechanics vol 23 no 2 pp203ndash210 2001

[16] C Cai H Zheng M Khan and K HungModeling of MaterialDamping Properties in ANSYS Defense Systems Division Insti-tute of High Performance Computing 2002

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

14 Shock and Vibration

005

0

minus0050 2 4 6 8 10 12

0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12

0 2 4 6 8 10 120 2 4 6 8 10 12

0 2 4 6 8 10 12 0 2 4 6 8 10 120 2 4 6 8 10 12

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at732

Hz (

mm

)

0

008

006

004

002

minus002

minus004

minus006

minus008

0

008

006

004

002

minus002

minus004

minus006

minus008

0

008

006

004

002

minus002

minus004

minus006

minus008

0

015

01

005

minus005

minus01

01

005

0

minus005

minus01

2

15

1

05

0

minus05

minus1

minus15

minus2

03

02

01

0

minus01

minus02

03

02

01

0

minus01

minus02

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at752

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at772

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at792

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at812

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at832

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at852

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at872

Hz (

mm

)

Expe

rimen

tal t

ip-d

ispla

cem

ent f

rom

tran

sient

to st

eady

at892

Hz (

mm

)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Time (s) Time (s) Time (s)

Figure 13 Experimental displacement responses at the position of 119909 = 1204mm from transient to steady state

12

1

08

06

04

02

0120 100 80 60 40 20

78

80

82

84Theo

retic

al st

eady

disp

lace

men

t (m

m)

Position at the beam (mm)Freq

uency

(Hz)

Figure 14 Theoretical steady-displacement responses versus posi-tions and frequencies with mode 119880(2)

1(119909)

improvements of the analytical methods for calculating themode shapes of the cantilever beam with a tip-mass anda base excitation In addition the amplitudes of the modeshapes decreased with the increasing mass value of the tip-mass when the moment of inertia and offset were constantsThe amplitude of the first-order mode shape increased but

12

1

08

06

04

02

0120 100 80 60 40 20

78

80

82

84Theo

retic

al st

eady

disp

lace

men

t (m

m)

Position at the beam (mm)Freq

uency

(Hz)

Figure 15 Theoretical steady-displacement responses versus posi-tions and frequencies with mode 119880(3)

1(119909)

the other nine mode shapes decreased with the increasingmoment of inertia when the mass and offset were set asconstants Conversely the amplitude of the first mode shapedecreased but the other ninemode shapes increased with theincreasing offset distance under the condition that the massand moment of inertia were constants Globally the effect

Shock and Vibration 15

12

1

08

06

04

02

0

Expe

rimen

tal s

tead

y di

spla

cem

ent (

mm

)

120 100 80 60 40 20Position at the beam (mm)

78

80

82

84Freq

uency

(Hz)

Figure 16 Experimental steady-displacement responses versuspositions and frequencies

from varying combinations of the mass and moment inertiaof the tip-mass and the offset distance for the lower-ordermode shapes was stronger than for the higher ones

Furthermore it is clearly noticed that the mass of tip-mass can reduce the values of the natural frequencies andthe increasing mass value gave the decreasing frequencieswhen the moment of inertia of the tip-mass and the offsetdistance were set as constants Similarly the moment ofinertia can further decrease the natural frequencies and thehigher moment of inertia resulted in the lower frequencieswhen the mass and the offset were constants Differently theoffset distance can reduce the first-order natural frequenciesbut increase the other nine order frequencies and the higheroffset distance led to the lower first-order natural frequenciesbut produced the higher other nine order frequencies whenthe mass and the moment of inertia were set to constant Inglobal view the influence from the varying mass momentof inertia and offset distance for the lower-order naturalfrequencies was larger than for the higher-order ones weatherthis influence was negative or positive

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the Natural Science Foundationsof China (no 51275380) and the Scientific and TechnologicalProject of Shaanxi Province (no 2012K06-36) The authorswould like to thank the reviewers for their valuable commentson this paper

References

[1] P A A Laura J L Pombo and E A Susemihl ldquoA note on thevibrations of a clamped-free beam with a mass at the free endrdquoJournal of Sound and Vibration vol 37 no 2 pp 161ndash168 1974

[2] B Rama Bhat andHWagner ldquoNatural frequencies of a uniformcantilever with a tip mass slender in the axial directionrdquo Journalof Sound and Vibration vol 45 no 2 pp 304ndash307 1976

[3] C W S To ldquoVibration of a cantilever beam with a baseexcitation and tip massrdquo Journal of Sound and Vibration vol83 no 4 pp 445ndash460 1982

[4] P A A Laura ldquoComments on lsquoVibration of a cantilever beamwith a base excitation and tip massrsquordquo Journal of Sound andVibration vol 88 no 4 p 569 1983

[5] C W S To ldquoDiscussion on lsquovibration of a cantilever beam witha base excitation and tip massrsquordquo Journal of Sound and Vibrationvol 97 no 2 pp 349ndash351 1984

[6] R G Jacquot ldquoFurther comments on lsquoVibration of a cantileverbeam with a base excitation and tip massrsquordquo Journal of Sound andVibration vol 93 no 2 pp 312ndash313 1984

[7] S AMousavi Lajimi andG R Heppler ldquoComments on lsquonaturalfrequencies of a uniform cantilever with a tip mass slender inthe axial directionrsquordquo Journal of Sound and Vibration vol 331 no12 pp 2964ndash2968 2012

[8] R B Bhat ldquoAuthors response to the comments on lsquonaturalfrequencies of a uniform cantilever with a tip mass slender inthe axial directionrsquordquo Journal of Sound and Vibration vol 331 no12 pp 2962ndash2963 2012

[9] E Esmailzadeh and G Nakhaie-Jazar ldquoPeriodic behavior ofa cantilever beam with end mass subjected to harmonic baseexcitationrdquo International Journal of Non-Linear Mechanics vol33 no 4 pp 567ndash577 1998

[10] H Abramovich and O Hamburger ldquoVibration of a cantilevertimoshenko beam with a tip massrdquo Journal of Sound andVibration vol 148 no 1 pp 162ndash170 1991

[11] B Pratiher ldquoVibration control of a transversely excited can-tilever beam with tip massrdquo Archive of Applied Mechanics vol82 no 1 pp 31ndash42 2012

[12] M Kim M Hoegen J Dugundji and B L Wardle ldquoModelingand experimental verification of proof mass effects on vibrationenergy harvester performancerdquo Smart Materials and Structuresvol 19 Article ID 045023 2010

[13] Y T Li and R Wong ldquoIntegral and series representations ofthe Dirac delta functionrdquo Communications on Pure and AppliedAnalysis vol 7 no 2 pp 229ndash247 2008

[14] M Al-Gwaiz Sturm-LiouvilleTheory and its Applications chap-ter 3 Springer London UK 2008

[15] F Bao-lian ldquoGeneralized reciprocal theorems and their appli-cationsrdquo Applied Mathematics and Mechanics vol 23 no 2 pp203ndash210 2001

[16] C Cai H Zheng M Khan and K HungModeling of MaterialDamping Properties in ANSYS Defense Systems Division Insti-tute of High Performance Computing 2002

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 15

12

1

08

06

04

02

0

Expe

rimen

tal s

tead

y di

spla

cem

ent (

mm

)

120 100 80 60 40 20Position at the beam (mm)

78

80

82

84Freq

uency

(Hz)

Figure 16 Experimental steady-displacement responses versuspositions and frequencies

from varying combinations of the mass and moment inertiaof the tip-mass and the offset distance for the lower-ordermode shapes was stronger than for the higher ones

Furthermore it is clearly noticed that the mass of tip-mass can reduce the values of the natural frequencies andthe increasing mass value gave the decreasing frequencieswhen the moment of inertia of the tip-mass and the offsetdistance were set as constants Similarly the moment ofinertia can further decrease the natural frequencies and thehigher moment of inertia resulted in the lower frequencieswhen the mass and the offset were constants Differently theoffset distance can reduce the first-order natural frequenciesbut increase the other nine order frequencies and the higheroffset distance led to the lower first-order natural frequenciesbut produced the higher other nine order frequencies whenthe mass and the moment of inertia were set to constant Inglobal view the influence from the varying mass momentof inertia and offset distance for the lower-order naturalfrequencies was larger than for the higher-order ones weatherthis influence was negative or positive

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the Natural Science Foundationsof China (no 51275380) and the Scientific and TechnologicalProject of Shaanxi Province (no 2012K06-36) The authorswould like to thank the reviewers for their valuable commentson this paper

References

[1] P A A Laura J L Pombo and E A Susemihl ldquoA note on thevibrations of a clamped-free beam with a mass at the free endrdquoJournal of Sound and Vibration vol 37 no 2 pp 161ndash168 1974

[2] B Rama Bhat andHWagner ldquoNatural frequencies of a uniformcantilever with a tip mass slender in the axial directionrdquo Journalof Sound and Vibration vol 45 no 2 pp 304ndash307 1976

[3] C W S To ldquoVibration of a cantilever beam with a baseexcitation and tip massrdquo Journal of Sound and Vibration vol83 no 4 pp 445ndash460 1982

[4] P A A Laura ldquoComments on lsquoVibration of a cantilever beamwith a base excitation and tip massrsquordquo Journal of Sound andVibration vol 88 no 4 p 569 1983

[5] C W S To ldquoDiscussion on lsquovibration of a cantilever beam witha base excitation and tip massrsquordquo Journal of Sound and Vibrationvol 97 no 2 pp 349ndash351 1984

[6] R G Jacquot ldquoFurther comments on lsquoVibration of a cantileverbeam with a base excitation and tip massrsquordquo Journal of Sound andVibration vol 93 no 2 pp 312ndash313 1984

[7] S AMousavi Lajimi andG R Heppler ldquoComments on lsquonaturalfrequencies of a uniform cantilever with a tip mass slender inthe axial directionrsquordquo Journal of Sound and Vibration vol 331 no12 pp 2964ndash2968 2012

[8] R B Bhat ldquoAuthors response to the comments on lsquonaturalfrequencies of a uniform cantilever with a tip mass slender inthe axial directionrsquordquo Journal of Sound and Vibration vol 331 no12 pp 2962ndash2963 2012

[9] E Esmailzadeh and G Nakhaie-Jazar ldquoPeriodic behavior ofa cantilever beam with end mass subjected to harmonic baseexcitationrdquo International Journal of Non-Linear Mechanics vol33 no 4 pp 567ndash577 1998

[10] H Abramovich and O Hamburger ldquoVibration of a cantilevertimoshenko beam with a tip massrdquo Journal of Sound andVibration vol 148 no 1 pp 162ndash170 1991

[11] B Pratiher ldquoVibration control of a transversely excited can-tilever beam with tip massrdquo Archive of Applied Mechanics vol82 no 1 pp 31ndash42 2012

[12] M Kim M Hoegen J Dugundji and B L Wardle ldquoModelingand experimental verification of proof mass effects on vibrationenergy harvester performancerdquo Smart Materials and Structuresvol 19 Article ID 045023 2010

[13] Y T Li and R Wong ldquoIntegral and series representations ofthe Dirac delta functionrdquo Communications on Pure and AppliedAnalysis vol 7 no 2 pp 229ndash247 2008

[14] M Al-Gwaiz Sturm-LiouvilleTheory and its Applications chap-ter 3 Springer London UK 2008

[15] F Bao-lian ldquoGeneralized reciprocal theorems and their appli-cationsrdquo Applied Mathematics and Mechanics vol 23 no 2 pp203ndash210 2001

[16] C Cai H Zheng M Khan and K HungModeling of MaterialDamping Properties in ANSYS Defense Systems Division Insti-tute of High Performance Computing 2002

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

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RoboticsJournal of

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Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of